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← Auctioning Off Household Goods During the Regency Era, Part 2
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The "Skinny" on Abdicating a Title During the Regency Era
Posted on August 29, 2022 by Regina Jeffers
Many times in Regency-based novels we have the situation where for one reason or another, the hero refused the title he has inherited and "abdicates" his new peerage. The question is whether this is a viable plot line.
The answer is a bit more complicated that we might expect. Let us say we have an earl who wishes to abdicate his title. He would have the option of refusing the title, the properties, and the money, but he would still technically be the earl until he dies and another secedes him. To have the full title and the honors accompanying it, the man would need to be confirmed before Parliament. In my story, "Courting Lord Whitmire," , there is a lengthy scene where Lord Andrew Whitmire appears before Parliament to claim the viscountcy after his father's death.] Parliament demands that the person making the claim to the title present evidence of his right to it. If the man wishes to be styled as an earl, he must claim the title. He does not need to send in the Writ of Summons to the House of Lords, and he can refuse to use the title, but someone must care for the property, and no one else can have the title while he is alive.
If he wished to claim the privileges of the peerage, which included: Peers had some special privileges. The main one was the right to sit in the House of Lords, unless they were Roman Catholic, a minor, a female or a lunatic. They could not be arrested for debts. They had to advance the peerage as an affirmative defense. They did not have to sit on juries. (This made sense as the House of Lords was in effect the supreme court and the last court of appeal). If arrested for a crime, they were allowed to be tried by the House of Peers. Their wives also claimed these privileges, except for sitting in the House of Lords. It was against the law to libel or slander a peer or to strike him. It was not until 1963 that anyone could walk away from a title.
When the Frederick Berkeley, 5th Earl of Berkley died, his oldest son applied for a Writ of Summons to the House of Lords. Berkeley and Mary Cole (who also passed under the name of Tudor), the daughter of a local publican and butcher, had seven sons and five daughters, but the disputed date of their marriage prevented their elder sons from succeeding as Earl of Berkeley and Baron Berkeley. The pair asserted their marriage had taken place on 30 March 1785, but the earliest ceremony of which there is incontrovertible proof was a wedding in Lambeth Church, Surrey, on 16 May 1796, at which date Mary was pregnant with their seventh child. Berkeley settled Berkeley Castle upon their eldest son, William FitzHardinge Berkeley, but William's attempt to assume his father's honours were disallowed by the House of Lords, who considered him illegitimate.
Therefore, the Committee on Privilege turned down the eldest's request, saying he and the other brothers born before 1795 were illegitimate, and the earldom had fallen to the 16-year-old born in 1796. Berkeley's titles devolved as a matter of law upon his fifth but first legitimate son, Thomas Morton Fitzhardinge Berkeley (1796–1882), but were never used by him and he did not take his seat in the House of Lords. Per his father's will, he would have lost his small inheritance had he disputed his eldest brother's claim to the titles. The boy was too young, for he had not reached his majority, to do anything about the matter, and his oldest brother and mother ran things. When he came of age, he still never put forth a claim to the earldom. However, he was, by right and law, the earl, so anything requiring the signature of the earl had to be signed by him. He signed responsibility over to his oldest brother, but the title itself went dormant until he died.
Somewhere around 1945, men succeeding to a peerage were allowed to disclaim it. The title went dormant during the man's life time. No one else could assume it. This was done mainly by men who had political power in the House of Commons and did not want to relinquish it. The current Duke of Marlborough cannot pass over his heir for a more somber, younger son. The heir can not be disinherited. If there is a living person who is next in line, the succession cannot be changed. Earlier the descent of the dukedom was changed because the Duke had no living son, and there was a slight chance of his having one. The succession was changed to allow his daughters to assume the title and then their sons. This was during the duke's life time and was an exception to a general rule. A man who is living and the lawful successor cannot have it taken from him except by being convicted of a crime. During the Regency, there was no way to disclaim a peerage except by not using it and not sending in a request for a seat in the House of Lords. Once a man was seated in the House of Lords under a title, the HOL would not take it back. Neither was he able to disclaim it.
For additional information, visit Nancy Mayer Regency Researcher's site for Succession When a Peer Dies and Introduction of a New Peer: Fees for Promotion and Introduction of a New Peer to the House of Lords.
For a good summary of what constitutes a "hereditary peer," visit https://en.wikipedia.org/wiki/Hereditary_peer
A man could be stripped of his title by the Crown if he committed treason, but not only would be tried and executed for his action, but his family would also be held "guilty." The University of Michigan's website refers to Blackstone's summary of the laws:
"Since High Treason was, and arguably remains, the most serious capital crime, testimony of two witnesses to the same overt act was required to convict, and the punishment in the Eighteenth century was severe. Blackstone states that 'the punishment of high treason in general is very solemn and terrible':
That the offender be drawn to the gallows, and not carried or walk: though usually (by connivance length ripened by humanity into law) a sledge or hurdle is allowed, to preserve the offender from the extreme torment of being dragged on the ground or pavement
That he be hanged by the neck and then cut down alive
That his entrails be taken out and burned, while he is yet alive
That his head be cut off
That his body be divided in four parts
That his head and quarters be at the King's disposal. [Blackstone, Wm., Knight. Chase, George, ed. Chase's Blackstone Commentaries on the Laws of England in Four Books. New York: Baker, Voorhis & Co., 1936. p889.]
"The punishment did not end with the personal suffering of the offender: the punishment extended to his or her family. The law states that a person who is found guilty of treason must also undergo "forfeiture" and "corruption of blood." In forfeiture, the person is force to give all their lands and property to the Crown. Corruption of blood prevents the person's immediate family and hereditary heirs from owning property or conducting business—in effect ruining the offender's family forever."
On the other hand, if the peer committed suicide, nothing happened to the title. The son inherited as usual. It would be a rare man of that time who did not want a title just because his father had disgraced it. He was not required to claim it, but he could not sit in the House of Lords if he did not. He could change his name either by sign manual, deed poll, or just by doing it. However, those are extreme measures, and he would be compounding the failure of his father by not attending to the estate, the workers, the servants and all the others who depend on the family in one way or another. [I use all this legal rigamarole in my upcoming book, The Heartless Earl, being released by Black Opal Books on October 16. The earl is accused of a crime that puts not only his life, but the earldom, in jeopardy.]
Obviously, we have had royalty abdicate in recent times. Edward VIII became king on his father's (George V) death in early 1936. However, he showed impatience with court protocol, and caused concern among politicians by his apparent disregard for established constitutional conventions. Only months into his reign, he caused a constitutional crisis by proposing to Wallis Simpson, an American who had divorced her first husband and was seeking a divorce from her second. The prime ministers of the United Kingdom and the Dominions opposed the marriage, arguing a divorced woman with two living ex-husbands was politically and socially unacceptable as a prospective queen consort. Additionally, such a marriage would have conflicted with Edward's status as the titular head of the Church of England, which, at the time, disapproved of remarriage after divorce if a former spouse was still alive. Edward knew the British government, led by Prime Minister Stanley Baldwin, would resign if the marriage went ahead, which could have forced a general election and would ruin his status as a politically neutral constitutional monarch. When it became apparent he could not marry Wallis and remain on the throne, Edward abdicated. He was succeeded by his younger brother, George VI.
After his abdication, he was created Duke of Windsor. Edward married Wallis in France on 3 June 1937, after her second divorce became final. Later that year, the couple toured Germany. During the Second World War, he was at first stationed with the British Military Mission to France, but after private accusations that he held Nazi sympathies, he was appointed Governor of the Bahamas. After the war, Edward spent the rest of his life in retirement in France. Edward and Wallis remained married until his death in 1972. [Edward VIII]
One final question that has likely occurred to some of you reading this: Is it possible for all lands to be lost save to pay debts? Or did lands always have to remain with the title?
It really depends on how the lands were acquired and what deeds, settlements, or entails control their disposition. While all of a man's property could be sold to cover debts, entailed properties and hereditaments [In law, a hereditament (from Latin hereditare, to inherit, from heres, heir) is any kind of property that can be inherited.] and such came under a special provision. Now, there were special rules pertaining to peers and debts. Land did not always go with the title though the family seat was usually tied to it. Titles and property could go in different directions and often did.
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Regina Jeffers is the award-winning author of Austenesque, Regency and historical romantic suspense.
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This entry was posted in Black Opal Books, British history, Church of England, customs and tradiitons, estates, family, Georgian England, Georgian Era, history, Inheritance, kings and queens, legacy, peerage, real life tales, Regency era, research, titles of aristocracy, tradtions and tagged abdication, British history, Duke of Windsor, Earl of Berkeley, estates, family, inheritance, kings and queens, last will & testament, law, legacy, peerage, royalty, titles of the aristocracy. Bookmark the permalink.
2 Responses to The "Skinny" on Abdicating a Title During the Regency Era
Elaine Mitrovich says:
Regina- what an excellent explanation of a complicated subject. Very enlightening.
reginajeffers says:
I am glad you found it helpful, Elaine.
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WEDNESDAY, March 13 (HealthDay News) -- An experimental blood test could help show whether women with advanced breast cancer are responding to treatment, a preliminary study suggests.
The test detects abnormal DNA from tumor cells circulating in the blood. And the new findings, reported in the March 14 issue of the New England Journal of Medicine, hint that it could outperform existing blood tests at gauging some women's response to treatment for metastatic breast cancer.
That's an advanced form of breast cancer, where tumors have spread to other parts of the body -- most often the bones, lungs, liver or brain. There is no cure, but chemotherapy, hormonal therapy or other treatments can slow disease progression and ease symptoms.
The sooner doctors can tell whether the treatment is working, the better. That helps women avoid the side effects of an ineffective therapy, and may enable them to switch to a better one.
Right now, doctors monitor metastatic breast cancer with the help of imaging tests, such as CT scans. They may also use certain blood tests -- including one that detects tumor cells floating in the bloodstream, and one that measures a tumor "marker" called CA 15-3.
But imaging does not tell the whole story, and it can expose women to significant doses of radiation. The blood tests also have limitations and are not routinely used.
"Practically speaking, there's a huge need for novel methods" of monitoring women, said Dr. Yuan Yuan, an assistant professor of medical oncology at City of Hope cancer center in Duarte, Calif.
For the new study, researchers at the University of Cambridge in England took blood samples from 30 women being treated for metastatic breast cancer and having standard imaging tests. They found that the tumor DNA test performed better than either the CA 15-3 or the tumor cell test when it came to estimating the women's treatment response.
Of 20 women the researchers were able to follow for more than 100 days, 19 showed cancer progression on their CT scans. And 17 of them had shown rising tumor DNA levels. In contrast, only seven had a rising number of tumor cells, while nine had an increase in CA 15-3 levels.
For 10 of those 19 women, tumor DNA was on the rise an average of five months before CT scans showed their cancer was progressing.
"The take-home message is that circulating tumor DNA is a better monitoring biomarker than the existing Food and Drug Administration-approved ones," said senior researcher Dr. Carlos Caldas.
It all suggests that the test could help in monitoring women's treatment response, said Yuan, who was not involved in the study.
But while she said the findings are "exciting," she also stressed that a lot more work needs to be done.
"This is nowhere near being ready for clinical practice," Yuan said. "But this is one direction we're heading in."
There are other tests being developed for monitoring women with breast cancer, Yuan noted. One is a test that looks for abnormalities in DNA "copy number." A recent preliminary study found that this approach might help predict some women's risk of a breast cancer recurrence.
And researchers are still studying existing tests to see how they can best be used. The blood test that detects tumor cells -- sold in the United States as the CellSearch system -- can be used to help monitor women in treatment for metastatic breast cancer. In general, a higher number of tumor cells means a quicker progression.
But for now, professional guidelines do not recommend that doctors routinely use the test because its ultimate usefulness is still unclear, said Dr. Anthony Lucci, a surgical oncologist at the University of Texas M.D. Anderson Cancer Center in Houston.
The new findings suggest that the tumor DNA test is more sensitive than the existing tumor cell test, said Lucci, who was not involved in the research.
He said that in the future, it might be helpful in monitoring women with metastatic cancer or in helping to spot a breast cancer recurrence earlier.
Earlier detection of recurrences is the big hope, said Dr. Jorge Reis-Filho, an attending pathologist at Memorial Sloan-Kettering Cancer Center in New York City. "If changes in DNA happen before changes are seen in imaging," he said, "that could help us be more proactive in treatment."
But, Reis-Filho stressed, that's "crystal-ball gazing" for now.
Lucci said any real-world use of tumor DNA testing is a long way off.
"Number one, we need larger studies to confirm these findings," he said. But beyond that, researchers need to figure out how to do such DNA testing in a simpler, cheaper way, Lucci added.
"Currently, this would be way too expensive and time-consuming," he said. Only some academic cancer centers would have the resources to do this kind of testing as it stands, Lucci noted.


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Day 01: Arrive Leh Arrival Leh Kushok Bakula Airport (This must be one of the Most Sensational Flights in the World. On a clear day from one side of the aircraft can be seen in the distance the peaks of K2, Nanga Parbat, Gasherbrum and on the other side of the aircraft, so close that you feel you could reach out and touch it, is the Nun Kun massif.) Upon arrival you will met by MT representative and transfer to Hotel for Check in. Complete day for rest and leisure to acclimatize followed by Welcome tea or Coffee at the Hotel. Evening visit to Leh Palace, Shanti Stupa & Leh Market. Overnight at Hotel.
Day 02: Leh - Excursion To Sham Valley (50 Kms / 03-04 Hrs ) After breakfast you drive downstream along the River Indus on Leh – Kargil Highway. And start our sightseeing with Hall Of Fame (Museum Constructed by India Army) near Leh is worth a visit for every Indian, and proceed further to Gurudwara Patthar Sahib at an altitude of 3600 Mtrs about 22 Kms from Leh was constructed in memory of Guru Nanak Dev ji Nestled deep in the Himalayas, A drive of another 4 km took us to Magnetic Hill which defies the law of gravity. As you move ahead from Magnetic Hills, just before Nimmu Village, there comes a famous Confluence Of Two Rivers Indus River coming from Tibet (left) and Zanskar River coming from Zanskar Valley (right). Zanskar River is known for its famous Chadar Trek. Later we drive back to Leh. Dinner and overnight at Hotel in Leh.
Day 03: leh–nubra valley via khardung la 18,390 ft. (120 kms / 4 – 5 hrs) Post breakfast we drive to Nubra Valley. The road journey from Leh to Nubra Valley passes over Khardung La (The Highest Motorable Road In The World) at 5,602 Mtrs / 18,390 Ft, around 39 km from Leh. From the pass, one can see all the way south over the Indus valley to seemingly endless peaks and ridges of the Zanskar range, and north to the giants of the Saser massif. Nubra Valley is popularly known as Ldumra or the valley of flowers. It is situated in the north of Ladakh between the Karakoram and Ladakh ranges of the Himalayas. The average altitude of the valley is 10,000 Ft. above sea level.
Day 04: Nubra Valley To Pangong Lake (170 Kms / 7 Hrs) After an early breakfast we check out from Camp or hotel and drive towards Hunder where you have opportunity to ride on a bacterian camel (double humped) amidst the sand dunes of this high altitude desert with snowcapped mountains on the horizon. Between Hunder and Diskit you come across sand dunes besides the road. Seen in isolation you could well be amongst the sand dunes of the Thar Desert in Rajasthan.
Later we drive towards Pangong Tso via Agham & Shyok Valley. Some of the road patches in areas are extremely tough (towards nubra side) in terms of road conditions but no doubt the journey is very scenic! Later after crossing this adventurous part we drive further towards Durbuk & Tangtse for Pangong you pass through few small villages of Changthang and finally you can have a sudden view of the Pangong lake situated at 14,000 feet (4,267 m). This famous blue brackish Lake of Pangong is 5/6 Kms wide and over 133 Kms long with half of its running the other side of the "Indo China Border". One rarely feels so close to nature and environment and the scenery is unforgettable. After the success of 3 Idiots, this beautiful natural location has attracted more tourists from all over the world. Upon arrival we check in at our Camp Near Pangong lake. Dinner & Overnight at the Camp.
Day 06: Depart Leh (Fly Out) Tour concludes.Thanks you for your patronage and ensures a punctual transfer to Domestic Airport to catch the flight for your onward Destination.
Leh - Excursion To Sham Valley (50 Kms / 03-04 Hrs ) After breakfast you drive downstream along the River Indus on Leh – Kargil Highway. And start our sightseeing with Hall Of Fame (Museum Constructed by India Army) near Leh is worth a visit for every Indian, and proceed further to Gurudwara Patthar Sahib at an altitude of 3600 Mtrs about 22 Kms from Leh was constructed in memory of Guru Nanak Dev ji Nestled deep in the Himalayas, A drive of another 4 km took us to Magnetic Hill which defies the law of gravity. As you move ahead from Magnetic Hills, just before Nimmu Village, there comes a famous Confluence Of Two Rivers Indus River coming from Tibet (left) and Zanskar River coming from Zanskar Valley (right). Zanskar River is known for its famous Chadar Trek. Later we drive back to Leh. Dinner and overnight at Hotel in Leh.
Depart Leh (Fly Out) Tour concludes. thanks you for your patronage and ensures a punctual transfer to Domestic Airport to catch the flight for your onward Destination.
Pangong Sarai boast of having the best located camp site at the world famous -turquoise blue Pangong Lake. At a distance of 05 minutes walk from the shore of the lake in Maan Village, we have 32 Swiss tents with attached bath.


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We had 5 HS students participate in an area chess tournament at Natalia High School this past Saturday. The students had a good time, and Lytle took the first place team award and 3rd place individual award (Joel Guerrero). We will be hosting another tournament in Lytle on March 2nd.


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text:
In Marrakech, Nervously Waiting to Learn Whether America Will Take a 'Lead Pipe' to the Paris…
Pacific Standard Staff
Regardless what happens in today's U.S. election, leaders at COP22 in Marrakech stress that the world is moving forward—but, in private, a lot of people here express concern.
By Eric J. Lyman
Some very powerful people, sitting in Marrakech, possibly thinking about Donald Trump. (Photo: Fadel Senna/AFP/Getty Images)
It isn't news that much of the world is eagerly awaiting results of Tuesday's presidential election in the United States. But in one corner of Morocco—Marrakech, the Red City, serving as home to the latest round of United Nations climate negotiations—the wait is particularly poignant.
The 22nd Conference of the Parties, best known as COP22, got underway Monday, and, in almost every COP22 press conference before or after the opening, there was a question or two about what might happen if Donald Trump — the Republican nominee who calls climate change a "hoax" and says he would "cancel" last year's Paris Agreement if elected— prevails over rival Hillary Clinton, a strong supporter of climate action.
A round of nervous giggles often accompanies the Trump questions in Marrakech, but it's clear that visiting delegates and observers will laugh much easier if Trump is defeated.
At last year's summit in the French capital, delegates from nearly 200 countries finalized the Paris Agreement, the world's first global climate pact. This year, countries are starting work on the "rulebook" for implementation of the Paris deal. Uncertainty coming from U.S. election is, as one climate activist calls it, an "unwelcome distraction."
Go back 16 years and there is a kind of uneasy precedent to the current situation. That year, delegates gathered at COP6 in the Hague to start work on the "rulebook" for the 1997 Kyoto Protocol. Those talks started a week after the U.S. elections, but, because of the famously contested results in Florida, it took more than a month before it was clear who would be the next president.
At the time, some delegations in the Hague wanted to craft a deal that the lame-duck Bill Clinton administration could accept; others wanted to hold out for stronger terms that would land on the desk of the next U.S. president. The talks went into overtime but remained stalemated. Negotiations were "suspended" rather than "closed," and they reconvened eight months later in Bonn, Germany — four months after newly installed President George W. Bush announced that the U.S. would reject the Kyoto treaty.
The Kyoto Protocol included obligations to reduce emissions for three-dozen industrialized countries, but with the U.S. — the world's largest polluter at that time — on the sidelines, it never had the far-reaching impact its architects intended.
In Marrakech, the conference's leadership insists that history will not be repeated with the Paris deal, the successor to the Kyoto Protocol, even if Trump wins the election.
"I think we should not over-emphasize the importance of Donald Trump," says Laurence Tubiana, France's climate change ambassador, who adds that it would be a "shock" to her if Trump is victorious.
"But even if that happened, even if Donald Trump wins," she continues, "we would all wake up Wednesday morning and still say 'We will stick to the Paris Agreement.'"
Other observers say climate-related fears from a possible Trump presidency might be overblown.
"If Trump somehow wins, the main incentive for the U.S. to remain in the process won't be something we do here, some rule that might get passed," says Alden Meyer, director of strategy and policy for the U.S.-based Union of Concerned Scientists. "The consequences for any country that might choose to disregard the Paris Agreement will come in the form of pressure from other countries in the areas that matter most to them: trade, energy, whatever the case may be. It would hit them where it hurts."
In the formal national statements during the opening plenaries, Trump was never mentioned by name, a nod to a tradition for multi-lateral negotiations to stay clear of the internal affairs of individual countries. But speaker after speaker nonetheless addressed Trump indirectly in stressing the need for increased international cooperation to confront climate risks, the incontrovertibility of climate science, and the obligation that large, wealthy countries have to play a leadership role in the process.
But in the corridors and meeting rooms of the Bab Ighli facility hosting the COP22 talks, there is no prohibition against discussing the New York businessman in more direct terms.
"We have been delicately constructing a mechanism that will help us confront this global problem, piece by piece, over the course of years," says one veteran European Union negotiator, who asks not to be named. "On Tuesday, in the U.S., they will vote on whether to elect a man who says he wants to take a hammer, a lead pipe, and a chain saw to that mechanism."
The delegate concludes: "So, yes, we may be a little nervous."
By the time the polls close in the U.S., it will be well into the early hours of Wednesday morning local time in Marrakech. But that will not stop dozens of journalists, delegates, activists, and other observers from gathering in Marrakech's Royal Mirage Deluxe Hotel to anxiously watch the election results roll in.
"We'll either have something to celebrate, or we'll all be able to talk each other down from the ledge," one of the e-mailed invitations to the event reads.
Pacific Standard publishes stories that matter, stories that, by virtue of their ideas and craft, are capable of creating a better and more just society. With a methodology that mixes reporting and narrative journalism with peer-reviewed research, we are fiercely committed to covering social and environmental justice.
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With AI becoming the trend in the tech industry, many tech companies large and small as well as investors, have been piling up their money to advance and make use of AI.
Among big tech companies that are into AIs, is Nvidia, the manufacturer of computer chips.
Here, the company's co-founder and CEO, Jensen Huang, believes that "AI is going to infuse all of software."
Nvidia itself has seen an increasing revenue since it started creating hardware dedicated to machine-learning algorithms for use cases such as autonomous cars. At the company's annual developer conference in San Jose, California, Huang explained how the machine-learning revolution is just starting.
"Software is eating the world, but AI is going to eat software," he said.
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"We're going to, for the first time, see the transmission of automation the way we're seeing the transmission and wireless broadcast of information for the very first time. I'm going to be able to send you automation. Send you a little automation by email."
However, the AI supremacy is a race for someone to conquer.
While AI is going to eat software, it still needs software to run, even if it's designed specifically for AI. As computers don't run on flesh and blood, it needs something to contain it. And that wouldn't be on thoughts. So here, Huang predicted that the future when AI can really work on its own, is still a long way from happening.
Two industries that are affected the most, include the automotive industry and healthcare.
"We can't ignore the impact of a technology that brings 10 or 1,000 times better results. I have confidence that reasonable minds will realize the benefits of this technology and put it in the hands of doctors and clinicians and radiologists so that they can do better work."
"The ability for AI to transform industries is well understood now." So indeed, the machine-learning revolution is starting.


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text:
\section{Introduction}

The problem of constructing polynomials having all coefficients in the set $\{-1,1\}$ (frequently called \emph{Littlewood polynomials}) with small $L^\alpha$ norm on the complex unit circle arises naturally in complex analysis~\cite{Lit1966},~\cite{Lit1968},~\cite{Bor2002},~\cite{Erd2002}, condensed matter physics~\cite{Ber1987}, and the design of sequences for communications devices~\cite{Gol1972},~\cite{BeeClaHer1985}.
\par
Recall that, for $1\le\alpha<\infty$, the $L^\alpha$ norm on the unit circle of a polynomial $f\in\mathbb{C}[z]$ is
\[
\norm{f}_\alpha=\left(\frac{1}{2\pi} \int_0^{2 \pi} \bigabs{f(e^{i\phi})}^\alpha \,d\phi\right)^{1/\alpha}.
\]
The $L^4$ norm has received particular attention because it is easier to calculate than most other $L^\alpha$ norms. Specifically, the $L^4$ norm of $f\in\mathbb{C}[z]$ is exactly the sum of the squared magnitudes of the coefficients of $f(z)\overline{f(z^{-1})}$. It is customary (see~\cite{Bor2002}, for example) to measure the smallness of the $L^4$ norm of a polynomial $f$ by its \emph{merit factor} $F(f)$, defined by
\[
F(f)=\frac{\norm{f}_2^4}{\norm{f}_4^4-\norm{f}_2^4},
\]
provided that the denominator is nonzero. Note that, if $f$ is a Littlewood polynomial of degree $n-1$, then $\norm{f}_2=\sqrt{n}$ and so a large merit factor means that the $L^4$ norm is small.
\par
We note that Golay's original equivalent definition~\cite{Gol1972} of the merit factor involves the aperiodic autocorrelations (which are precisely the coefficients of $f(z)\overline{f(z^{-1})}$) of the sequence formed by the coefficients of~$f$.
\par
Besides continuous progress on the merit factor problem in the last fifty years (see~\cite{Jed2005},~\cite{Hoh2006},~\cite{BorFerKna2008} for surveys and~\cite{JedKatSch2013} for a brief review of more recent work), modulo generalisations and variations, only three nontrivial families of Littlewood polynomials are known, for which we can compute the asymptotic merit factor. These are: Rudin-Shapiro polynomials and polynomials whose coefficients are derived either from multiplicative or additive characters of finite fields. As shown by Littlewood~\cite{Lit1968}, Rudin-Shapiro polynomials are constructed recursively in such a way that their merit factor satisfies a simple recurrence, which gives an asymptotic value of $3$. The largest asymptotic merit factors that have been obtained from the other two families equal cubic algebraic numbers $6.342061\dots$ and $3.342065\dots$, respectively~\cite{JedKatSch2013a},~\cite{JedKatSch2013} (see Corollary~\ref{cor:second_order} and Theorem~\ref{thm:gmw} in this paper
for precise statements).
\par
The latter polynomials are closely related to classical difference sets, namely Paley and Singer difference sets. Recall that a \emph{difference set} with parameters $(n,k,\lambda)$ is a $k$-subset $D$ of a finite group $G$ of order $n$ such that the $k(k-1)$ nonzero differences of elements in $D$ hit every nonzero element of~$G$ exactly $\lambda$ times (so that $k(k-1)=\lambda(n-1)$). We are interested in the case that the group $G$ is cyclic. In this case, we fix a generator $\theta$ of $G$ and associate with a subset $D$ of $G$ the Littlewood polynomial
\begin{equation}
f_{r,t}(z)=\sum_{j=0}^{t-1}\mathbbm{1}_D(\theta^{j+r})z^j,   \label{eqn:set_to_poly}
\end{equation}
where $r$ and $t$ are integers with $t\ge 0$ and
\[
\mathbbm{1}_D(y)=\begin{cases}
 1 & \text{for $y\in D$}\\
-1 & \text{for $y\in G\setminus D$}.
\end{cases}
\]
If $n$ is the group order, then $f_{0,n}$ captures the information about~$D$. We call this polynomial a \emph{characteristic polynomial} of~$D$ (which is unique up to the choice of $\theta$) and the polynomials $f_{r,n}$ \emph{shifted} characteristic polynomials.
\par
The results of this paper are mainly motivated by the 1991 paper of Jensen, Jensen, and H{\o}holdt~\cite{JenJenHoh1991}, in which the authors asked for the merit factor of polynomials derived from families of difference sets. It was shown in~\cite{JenJenHoh1991} that, among the known families of difference sets, only those with Hadamard parameters, namely
\[
(4h-1,\,2h-1,\,h-1)
\]
for a positive integer $h$, can give a nonzero asymptotic merit factor for their shifted characteristic polynomials. As of 1991, five families of such difference sets were known~\cite{JenJenHoh1991}:
\begin{enumerate}[(A)]
\item Paley difference sets,
\item Singer difference sets,
\item Twin-prime difference sets,
\item Gordon-Mills-Welch difference sets,
\item Hall difference sets.
\end{enumerate}
While in the first three cases, the asymptotic merit factors of the shifted characteristic polynomials have been determined in~\cite{HohJen1988} and~\cite{JenJenHoh1991} and those of the more general polynomials~\eqref{eqn:set_to_poly} in~\cite{JedKatSch2013a} and~\cite{JedKatSch2013}, the last two cases were left as open problems in~\cite{JenJenHoh1991}. More specifically, the asymptotic merit factor of the polynomials derived from a subclass of the Gordon-Mills-Welch difference sets is subject to a conjecture~\cite[Conjecture~7.1]{JedKatSch2013}. In this paper, we prove this conjecture and solve the problems concerning Gordon-Mills-Welch and Hall difference sets posed in~\cite{JenJenHoh1991} (see Theorem~\ref{thm:gmw} and Corollary~\ref{cor:sixth_order}, respectively). In addition, we obtain the asymptotic merit factor of polynomials related to a construction of Sidelnikov~\cite{Sid1969} (see also~\cite{LemCohEas1977}). This explains numerical observations in~\cite{HarYaz2010} and proves in the
affirmative~\cite[Conjecture~7.2]{JedKatSch2013}.
\par
In fact, the result for Hall difference sets arises from a much more general theorem concerning polynomials derived from cyclotomy (see Theorem~\ref{thm:mf_cyclotomic}). This result considers polynomials constructed from subsets of~$\mathbb{F}_p$ obtained by joining $m/2$ of the $m$ cyclotomic classes of (even) order $m$, where $m$ satisfies $p\equiv 1\pmod m$. The cases $m\in\{2,4,6\}$ are examined in detail. For $m=2$, we obtain the asymptotic merit factor of polynomials arising from Paley difference sets (see Corollary~\ref{cor:second_order}), which is the main result of~\cite{JedKatSch2013a}. For $m=4$, we obtain the asymptotic merit factor of polynomials arising from Ding-Helleseth-Lam almost difference sets~\cite{DinHelLam1999} (see Corollary~\ref{cor:fourth_order}). For $m=6$, we obtain, among other things, the asymptotic merit factor of polynomials arising from Hall difference sets (see Corollary~\ref{cor:sixth_order}).
\par
Some comments on our result for Gordon-Mills-Welch difference sets follow. In the cyclic case, such sets have paramaters
\[
(2^m-1,2^{m-1}-1,2^{m-2}-1),
\]
which are typically called \emph{Singer} parameters. The Gordon-Mills-Welch construction produces difference sets in a cyclic group $G$ of order $2^m-1$ from difference sets with Singer parameters in a subgroup of $G$. Hence this construction is very general and can in particular be iterated. Our result on Gordon-Mills-Welch difference sets (Theorem~\ref{thm:gmw}) requires no knowledge about the smaller difference sets that are used as building blocks. Thus Singer, Paley, or Hall difference sets (in groups whose order is a Mersenne number) can be used as building blocks. In addition, since 1991, further families of difference sets in cyclic groups with Singer parameters have been found:
\begin{enumerate}[(A)]
\setcounter{enumi}{5}
\item Maschietti difference sets~\cite{Mas1998},
\item Dillon-Dobbertin difference sets~\cite{DilDob2004},
\item No-Chung-Yun difference sets~\cite{DilDob2004}.
\end{enumerate}
Our results include the cases when these difference sets are used as building blocks in the Gordon-Mills-Welch construction. However we have not been able to determine the asymptotic merit factors of the polynomials associated with these difference sets themselves. We conjecture that they have the same behaviour as those of Singer and Gordon-Mills-Welch difference sets, given in Theorem~\ref{thm:gmw}.

\section{Results}
\label{sec:results}

To state our results, we require the function $\varphi_\nu:\mathbb{R}\times\mathbb{R}^{+}\to\mathbb{R}$, defined for real $\nu$ by
\begin{align*}
\frac{1}{\varphi_\nu(R,T)}=
1-\frac{2(1+\nu)T}{3}+4&\sum_{m\in\mathbb{N}}\max\bigg(0,1-\frac{m}{T}\bigg)^2\\
+\nu &\sum_{m\in\mathbb{Z}}\max\bigg(0,1-\biggabs{1+\frac{2R-m}{T}}\bigg)^2,
\end{align*}
where $\mathbb{N}$ is the set of positive integers. This function satisfies $\varphi_\nu(R,T)=\varphi_\nu(R+\tfrac{1}{2},T)$ on its entire domain. It will be useful to know the global maximum of $\varphi_\nu$ for certain values of $\nu$. The function $\varphi_1$ was maximised in~\cite[Corollary 3.2]{JedKatSch2013}. Using the same approach, we find that, for all $\nu\in[0,1]$, the global maximum of $\varphi_\nu(R,T)$ exists and equals the largest root of
\begin{multline*}
(\nu^4-2\nu^3-3\nu^2-50\nu+112)X^3+(12\nu^3+36\nu^2-18\nu-528)X^2\\ +(24\nu^2+282\nu+528)X-6\nu-48.
\end{multline*}
The global maximum is unique for $R\in[0,\tfrac{1}{2})$ and is attained when $T$ is the middle root of
\[
(2\nu+2)X^3-(6\nu+24)X+3\nu+24
\]
and $R=3/4-T/2$.
\par
We begin with stating our results for Gordon-Mills-Welch difference sets \cite{GorMilWel1962} whose ambient group is $\mathbb{F}_q^*$, where $q>2$ is a power of two\footnote{We note that~\cite{GorMilWel1962} defines more general difference sets, which are also called Gordon-Mills-Welch difference sets. However, the sets considered in this paper are the only ones with Hadamard parameters.}. Let~$\mathbb{F}_s$ be a proper subfield of $\mathbb{F}_q$ and let $A$ contain all elements $a\in\mathbb{F}_q$ with $\Tr_{q,s}(a)=1$, where $\Tr_{u,v}$ is the trace from $\mathbb{F}_u$ to $\mathbb{F}_v$. Let $B$ be a difference set in $\mathbb{F}_s^*$ with $\abs{B}=s/2$ (so that, for $s>2$, the complement of $B$ in $\mathbb{F}_s^*$ has Singer parameters). We also allow $s=2$, so that $B$ is a trivial difference set. A set of the form
\begin{equation}
\{ab:a\in A,b\in B\}   \label{eqn:gmw_construction}
\end{equation}
is a \emph{Gordon-Mills-Welch difference set} in $\mathbb{F}_q^*$ (whose complement has Singer parameters). They generalise the Singer difference sets, which arise for $s=2$. We have the following result for the asymptotic merit factor of polynomials obtained by Gordon-Mills-Welch difference sets.
\begin{theorem}
\label{thm:gmw}
Let $q>2$ be a power of two and let $f$ be a characteristic polynomial of a Gordon-Mills-Welch difference set in $\mathbb{F}_q^*$. Let $T>0$ be real. If $t/q\to T$, then $F(f_{r,t})\to \varphi_0(0,T)$ as $q\to\infty$.
\end{theorem}
\par
In the particular case of Singer difference sets, Theorem~\ref{thm:gmw} reduces to~\cite[Theorem~2.2~(i)]{JedKatSch2013}. The case that the Gordon-Mills-Welch difference sets in Theorem~\ref{thm:gmw} are of the form~\eqref{eqn:gmw_construction} when $B$ is a Singer difference set proves~\cite[Conjecture~7.1]{JedKatSch2013}\footnote{Conjecture~7.1 of \cite{JedKatSch2013} also involves ``negaperiodic'' and ``periodic'' extensions of the polynomials associated with Gordon-Mills-Welch difference sets. The corresponding assertions can be obtained as direct consequences of Proposition~\ref{pro:L_gmw} and~\cite[Theorem~4.2]{JedKatSch2013}, but are omitted here for the sake of simplicity.}.
\par
Next we consider subsets of $\mathbb{F}_q^*$ for an odd prime power $q$, which are related to a construction of Sidelnikov~\cite{Sid1969}. We call a set of the form
\begin{equation}
\{x\in\mathbb{F}_q^*:\text{$x+1$ is zero or a square in $\mathbb{F}_q^*$}\}   \label{eqn:def_sidelnikov}
\end{equation}
a \emph{Sidelnikov set} in $\mathbb{F}_q^*$. Such a set gives rise to a so-called almost difference set~\cite[Theorem~4]{AraDinHelKumMar2001}. We have the following result for the asymptotic merit factor of the associated polynomials, proving~\cite[Conjecture~7.2]{JedKatSch2013}.
\begin{theorem}
\label{thm:sidelnikov}
Let $q$ be an odd prime power and let $f$ be a characteristic polynomial of a Sidelnikov set in $\mathbb{F}_q^*$. Let $T>0$ be real. If $t/q\to T$, then $F(f_{r,t})\to \varphi_0(0,T)$ as $q\to\infty$.
\end{theorem}
\par
The maximum asymptotic merit factor that can be obtained in Theorems~\ref{thm:gmw} and~\ref{thm:sidelnikov} is $3.342065\dots$, the largest root of
\[
7X^3-33X^2+33X-3.
\]
\par
Next we construct Littlewood polynomials using cyclotomy. Let $m$ be a positive integer and let $p$ be a prime satisfying $p\equiv 1\pmod m$. Let $\omega$ be a fixed primitive element in $\mathbb{F}_p$. Let $C_0$ be the set of $m$-th powers in $\mathbb{F}_p^*$ and write $C_s=\omega^sC_0$ for $s\in\mathbb{Z}$. The sets $C_0,C_1,\dots,C_{m-1}$ partition $\mathbb{F}_p^*$ and are called the \emph{cyclotomic classes} of $\mathbb{F}_p$ of order $m$.
\par
We construct subsets $D$ of the additive group $\mathbb{F}_p$ by joining some of these classes. This method provides a rich source of difference sets (see~\cite{Jun1992} for a survey). We may take $1$ as a generator for $\mathbb{F}_p$, in which case the polynomials associated with $D$ are
\[
f_{r,t}(z)=\sum_{j=0}^{t-1}\mathbbm{1}_D(j+r)z^j
\]
and a characteristic polynomial is $f_{0,p}$ (this is no loss of generality; if the generator is $v$, then replace $D$ by $v^{-1}D$). It follows from~\cite[Theorem~2.1]{JenJenHoh1991} that the shifted characteristic polynomials associated with $D$ have a nonzero asymptotic merit factor only if $\abs{D}/p$ approaches $1/2$ as $p\to\infty$, thus $m$ must be even and $D$ must be a union of $m/2$ cyclotomic classes. Two families of difference sets arise in this way, namely the Paley difference sets for $m=2$ and the Hall difference sets for $m=6$~\cite[Theorem~2.2]{Hal1956}. If $D$ is a union of $m/2$ cyclotomic classes of order $m$, then $\abs{D}=(p-1)/2$, so if $D$ is a difference set, then it must have Hadamard parameters. Equivalently, $\abs{(D+u)\cap D}=(p-3)/4$ for every $u\in\mathbb{F}_p^*$. Our next theorem applies not only to such difference sets, but requires this condition to hold asymptotically (in a precise sense).
\begin{theorem}
\label{thm:mf_cyclotomic}
Let $m$ be an even positive integer and let $S$ be an $m/2$-element subset of $\{0,1,\dots,m-1\}$. Let $p$ take values in an infinite set of primes satisfying $p\equiv 1\pmod m$. Let $D$ be the union of the $m/2$ cyclotomic classes $C_s$ with $s\in S$ of $\mathbb{F}_p$ of order $m$ and suppose that, as $p\to\infty$,
\begin{equation}
\frac{(\log p)^3}{p^2}\sum_{u\in\mathbb{F}_p^*}\bigg(\bigabs{(D+u)\cap D}-\frac{p}{4}\bigg)^2\to 0.   \label{eqn:cond_Ru}
\end{equation}
Let $f$ be a characteristic polynomial of $D$ and let $R$ and $T>0$ be real. If $r/p\to R$ and $t/p\to T$, then the following hold as $p\to\infty$:
\begin{enumerate}[(i)]
\setlength{\itemsep}{1ex}
\item If $\frac{p-1}{m}$ is even for every $p$, then $F(f_{r,t})\to\varphi_1(R,T)$.
\item If $\frac{p-1}{m}$ is odd for every $p$, then $F(f_{r,t})\to\varphi_\nu(R,T)$, where $\nu=(\tfrac{4N}{m}-1)^2$ and
\[
N=\bigabs{\{(s,s')\in S\times S:s-s'=m/2\}}.
\]
\end{enumerate}
\end{theorem}
\par
Several remarks on Theorem~\ref{thm:mf_cyclotomic} follow. It is readily verified that $\nu$ in Theorem~\ref{thm:mf_cyclotomic} satisfies $\nu\in[0,1]$. The condition~\eqref{eqn:cond_Ru} is essentially necessary since
\[
\frac{1}{F(f)}\ge\frac{8}{p^2}\sum_{u\in \mathbb{F}_p^*}\bigg(\bigabs{(D+u)\cap D}-\frac{p-2}{4}\bigg)^2.
\]
This can be deduced from the proof of Theorem~\ref{thm:mf_cyclotomic} and the inequality
\[
\norm{f}_4^4\ge\frac{1}{2p}\sum_{k\in\mathbb{F}_p}\abs{f(e^{2\pi ik/p})}^4+\frac{p^2}{2},
\]
which can be obtained from~\cite[(2.3)]{HohJen1988} with an extra step involving the Cauchy-Schwarz inequality. In fact, this is a refinement of the Marcinkiewicz-Zygmund inequality~\cite[Chapter~X, Theorem~7.5]{Zyg1959} for the $L^4$ norm.
\par
The condition~\eqref{eqn:cond_Ru} can be checked using the cyclotomic numbers of order~$m$, which are the $m^2$ numbers
\[
\abs{(C_i+1)\cap C_j}
\]
for $0\le i,j<m$. These numbers are known explicitly for all even $m\le 20$ and for $m=24$ (see~\cite[p.~152]{BerEvaWil1998} for a list of references). Also note that the conclusion of Theorem~\ref{thm:mf_cyclotomic} remains unchanged if we replace $S$ by $h+S$ reduced modulo~$m$ for an integer $h$ (which changes $D$ to $\omega^hD$).
\par
We now consider in detail the cases $m\in\{2,4,6\}$ of Theorem~\ref{thm:mf_cyclotomic}. If $m=2$, then $D$ consists of either the squares or the nonsquares of $\mathbb{F}_p^*$. In both cases, $D$ is a Paley difference set for $p\equiv 3\pmod 4$. As remarked above, we can assume without loss of generality that $D$ is the set of squares in $\mathbb{F}_p^*$. Then we have (see~\cite[Theorem~2.2.2]{BerEvaWil1998}, for example)
\[
4\,\abs{(D+u)\cap D}=\begin{cases}
p-4-(-1)^{\frac{p-1}{2}} & \text{for $u$ a square in $\mathbb{F}_p^*$}\\[.5ex]
p-2+(-1)^{\frac{p-1}{2}} & \text{for $u$ a nonsquare in $\mathbb{F}_p^*$}.
\end{cases}
\]
Noting that $\nu=1$ for $m=2$, we obtain the following corollary, which is essentially the main result of~\cite{JedKatSch2013a} (see also~\cite[Theorem~2.1]{JedKatSch2013}).
\begin{corollary}
\label{cor:second_order}
Let $p$ take values in an infinite set of odd primes, let $D$ be either the set of squares or the set of nonsquares of $\mathbb{F}_p^*$ and let $f$ be a characteristic polynomial of $D$. Let $R$ and $T>0$ be real. If $r/p\to R$ and $t/p \to T$, then $F(f_{r,t})\to \varphi_1(R,T)$ as $p\to\infty$.
\end{corollary}
\par
We now look at the case $m=4$. Here, we only need to consider two cases for joining two cyclotomic classes of order four, namely $C_0\cup C_2$ and $C_0\cup C_1$. The first case brings us back to $m=2$. When $p$ is of the form $x^2+4$ for $x\in\mathbb{Z}$ and $(p-1)/4$ is odd, the second case gives rise to the Ding-Helleseth-Lam almost difference sets~\cite[Theorem~4]{DinHelLam1999}. By inspecting the cyclotomic numbers of order four (see Section~\ref{sec:residues}), we obtain the following.
\begin{corollary}
\label{cor:fourth_order}
Let $p$ take values in an infinite set of primes of the form $x^2+4y^2$ for $x,y\in\mathbb{Z}$ such that $y^2(\log p)^3/p\to 0$ as $p\to\infty$. Let $D$ be the union of two cyclotomic classes of $\mathbb{F}_p$ of order four and let $f$ be a characteristic polynomial of $D$. Let $R$ and $T>0$ be real. If $r/p \to R$ and $t/p \to T$, then $F(f_{r,t})\to \varphi_1(R,T)$ as $p\to\infty$.
\end{corollary}
\par
Recall from elementary number theory that primes of the form $x^2+4y^2$ for $x,y\in\mathbb{Z}$ are exactly the primes that are congruent to $1$ modulo $4$. It is also known~\cite{Col1993} that there are infinitely many primes satisfying the hypothesis of the corollary.
\par
The case $m=6$ is the first situation, where different limiting functions occur. In this case, there are four different sets $D$ to consider, namely
\begin{equation}
C_0\cup C_2\cup C_4,\quad C_0\cup C_1\cup C_2,\quad C_0\cup C_1\cup C_3,\quad C_0\cup C_1\cup C_4.   \label{eqn:sets_of_order_six}
\end{equation}
Again, the first set brings us back to $m=2$. When $p$ is of the form $x^2+27$ for $x\in\mathbb{Z}$ and $(p-1)/6$ is odd, then either the third or the fourth set in~\eqref{eqn:sets_of_order_six} gives rise to Hall difference sets~\cite{Hal1956} (the choice depends on the primitive element~$\omega$). We shall see that Theorem~\ref{thm:mf_cyclotomic} gives two possible limiting functions for the sixth cyclotomic classes, which is our motivation for the following definition. Let $D$ be a union of three cyclotomic classes of order six. If there is a $\gamma\in\mathbb{F}_p^*$ such that $\gamma D$ equals one of the first two sets in~\eqref{eqn:sets_of_order_six}, then we say that $D$ is of \emph{Paley type}. Otherwise, we say that $D$ is of \emph{Hall type}.
\begin{corollary}
\label{cor:sixth_order}
Let $p$ take values in an infinite set of primes of the form $x^2+27y^2$ for $x,y\in\mathbb{Z}$ such that $y^2(\log p)^3/p\to 0$ as $p\to\infty$. Let $D$ be the union of three cyclotomic classes of $\mathbb{F}_p$ of order six and let $f$ be a characteristic polynomial of $D$. Let $R$ and $T>0$ be real. If $r/p\to R$ and $t/p\to T$, then the following hold as $p\to\infty$:
\begin{enumerate}[(i)]
\setlength{\itemsep}{1ex}
\item If, for each $p$, $D$ is of Paley type or $\frac{p-1}{6}$ is even, then $F(f_{r,t})\to \varphi_1(R,T)$.
\item If, for each $p$, $D$ is of Hall type and $\frac{p-1}{6}$ is odd, then $F(f_{r,t})\to \varphi_{1/9}(R,T)$.
\end{enumerate}
\end{corollary}
\par
It is known that primes of the form $x^2+27y^2$ for $x,y\in\mathbb{Z}$ are exactly the primes $p$ for which $p\equiv 1\pmod 6$ and $2$ is a cube modulo $p$~\cite[Corollary~2.6.4]{BerEvaWil1998}. Again, it is also known~\cite{Col1993} that there are infinitely many primes satisfying the hypothesis of the corollary.
\par
The largest asymptotic merit factor that can be obtained in Corollaries~\ref{cor:second_order},~\ref{cor:fourth_order}, and~\ref{cor:sixth_order}~(i) is $6.342061\dots$, the largest root of
\[
29X^3-249X^2+417X-27,
\]
which equals the best known asymptotic value for Littlewood polynomials. The largest asymptotic merit factor that can be obtained in Corollary~\ref{cor:sixth_order}~(ii) is $3.518994\dots$, the largest root of
\[
349061X^3-1737153X^2+1835865X-159651.
\]
\par
It is also of interest to look at the case $T=1$ in our results, which concerns just the shifted characteristic polynomials, as considered in~\cite{HohJen1988} and~\cite{JenJenHoh1991} for Paley and Singer difference sets, respectively. Since
\[
\frac{1}{\varphi_\nu(R,1)}=\tfrac{1}{6}(2-\nu)+8\nu(R-\tfrac{1}{4})^2\quad\text{for $0\le R\le \tfrac{1}{2}$},
\]
the global maximum of $g_\nu(R,1)$ equals $6/(2-\nu)$. Hence, for $T=1$, Theorems~\ref{thm:gmw} and~\ref{thm:sidelnikov} give an asymptotic merit factor of $3$, Corollaries~\ref{cor:second_order},~\ref{cor:fourth_order}, and~\ref{cor:sixth_order}~(i) give a maximum asymptotic merit factor of $6$ and Corollary~\ref{cor:sixth_order}~(ii) gives a maximum asymptotic merit factor of $54/17$.
\par
We note that there are also ``negaperiodic'' and ``periodic'' versions of Theorem~\ref{thm:gmw}, Theorem~\ref{thm:mf_cyclotomic}, and its corollaries, as considered in~\cite{JedKatSch2013} (but not of Theorem~\ref{thm:sidelnikov} since in this case the characteristic polynomials have odd degree). These follow directly from our results and a generalisation of Theorem~\ref{thm:mf_from_L_1} in the vein of parts (ii) and (iii) of Theorems~4.1 and 4.2 in~\cite{JedKatSch2013}. We omit their statements for the sake of simplicity.
\par
We shall prove Theorems~\ref{thm:gmw} and~\ref{thm:sidelnikov} in Sections~\ref{sec:gmw} and~\ref{sec:sidelnikov}, respectively. Theorem~\ref{thm:mf_cyclotomic} and Corollaries~\ref{cor:fourth_order} and~\ref{cor:sixth_order} will be proved in Section~\ref{sec:residues}.

\section{Asymptotic merit factor calculation}

Let $f(z)=\sum_{j=0}^{n-1}a_jz^j$ be a Littlewood polynomial of degree $n-1$ and let~$r$ and $t$ be integers with $t\ge 0$. Define the polynomial
\[
f_{r,t}(z)=\sum_{j=0}^{t-1}a_{j+r}z^j,
\]
where we extend the definition of $a_j$ so that $a_{j+n}=a_j$ for all $j\in\mathbb{Z}$. Write $\epsilon_k=e^{2\pi ik/n}$. From~\cite{JedKatSch2013} it is known that $F(f_{r,t})$ depends only on the function $L_f:(\mathbb{Z}/n\mathbb{Z})^3\to\mathbb{Z}$, defined by
\[
L_f(a,b,c)=\frac{1}{n^3}\sum_{k\in\mathbb{Z}/n\mathbb{Z}}f(\epsilon_k)f(\epsilon_{k+a})\overline{f(\epsilon_{k+b})}\overline{f(\epsilon_{k+c})}.
\]
Define the functions $I_n,J_n:(\mathbb{Z}/n\mathbb{Z})^3\to\mathbb{Z}$ by
\begin{align*}
I_n(a,b,c)&=\begin{cases}
1 & \text{if ($c=a$ and $b=0$) or ($b=a$ and $c=0$),}\\
0 & \text{otherwise}
\end{cases}
\intertext{and}
J_n(a,b,c)&=\begin{cases}
1 & \text{if $a=0$ and $b=c\ne 0$,}\\
0 & \text{otherwise}
\end{cases}
\intertext{and, for even $n$, the function $K_n:(\mathbb{Z}/n\mathbb{Z})^3\to\mathbb{Z}$ by}
K_n(a,b,c)&=\begin{cases}
1 & \text{if $a=n/2$ and $b=c+n/2$ and $bc\ne 0$,}\\
0 & \text{otherwise}.
\end{cases}
\end{align*}
In order to prove Theorems~\ref{thm:gmw},~\ref{thm:sidelnikov}, and~\ref{thm:mf_cyclotomic} we shall show that the corresponding function $L_f$ is well approximated by either $I_n+\nu J_n$ for an appropriate real $\nu$ or by $I_n+K_n$ and then apply one of the following two theorems. Our first theorem is a slight generalisation of Theorems~4.1~(i) and~4.2~(i) of~\cite{JedKatSch2013}, which arise by setting $\nu=1$ and $\nu=0$, respectively. This theorem can be proved by applying straightforward modifications to the proof of~\cite[Theorem~4.1]{JedKatSch2013}.
\begin{theorem}
\label{thm:mf_from_L_1}
Let $\nu$ be a real number and let $n$ take values in an infinite set of positive integers. For each $n$, let $f$ be a Littlewood polynomial of degree $n-1$ and suppose that, as $n\to\infty$,
\[
(\log n)^3\max\limits_{a,b,c\in\mathbb{Z}/n\mathbb{Z}}\bigabs{L_f(a,b,c)-(I_n(a,b,c)+\nu J_n(a,b,c))}\to 0.
\]
Let $R$ and $T>0$ be real. If $r/n\to R$ and $t/n\to T$, then $F(f_{r,t})\to\varphi_{\nu}(R,T)$ as $n\to\infty$.
\end{theorem}
\par
There is a similar generalisation of parts (ii) and (iii) of Theorems~4.1 and~4.2 in~\cite{JedKatSch2013}, which we do not consider in this paper.
\par
Our second theorem is a more subtle modification of Theorem~4.1~(i) in~\cite{JedKatSch2013}. We include a proof that highlights the required modifications of the proof of \cite[Theorem~4.1~(i)]{JedKatSch2013}.
\begin{theorem}
\label{thm:mf_from_L_2}
Let $n$ take values in an infinite set of even positive integers. For each $n$, let $f$ be a Littlewood polynomial of degree $n-1$ and suppose that, as $n\to\infty$,
\begin{equation}
(\log n)^3\max\limits_{a,b,c\in\mathbb{Z}/n\mathbb{Z}}\bigabs{L_f(a,b,c)-(I_n(a,b,c)+ K_n(a,b,c))}\to 0.   \label{eqn:assumption_LIK}
\end{equation}
Let $T>0$ be real. If $t/n\to T$, then $F(f_{r,t})\to \varphi_0(0,T)$ as $n\to\infty$.
\end{theorem}
\begin{proof}
The first part of the proof is identical to that of~\cite[Theorem~4.1~(i)]{JedKatSch2013}, giving
\begin{equation}
\frac{1}{F(f_{r,t})}=-1+\frac{1}{t^2n}\;\sums{0\le j_1,j_2,j_3,j_4<t \\ j_1+j_2=j_3+j_4}\;\sum_{a,b,c\in\mathbb{Z}/n\mathbb{Z}}\;L_f(a,b,c)\,\epsilon_a^{-j_2-r}\epsilon_b^{j_3+r}\epsilon_c^{j_4+r},   \label{eqn:F_from_L}
\end{equation}
where $\epsilon_k=e^{2\pi ik/n}$. Write
\begin{equation}
L_f(a,b,c)=I_n(a,b,c)+K_n(a,b,c)+M_n(a,b,c),   \label{eqn:IplusK}
\end{equation}
where $M_n(a,b,c)$ is an error term, which can be controlled using~\eqref{eqn:assumption_LIK}. Consider three cases for the tuple $(a,b,c)\in\mathbb{Z}/n\mathbb{Z}$: (1) $c=a$ and $b=0$, (2) $a=b$ and $c=0$, and (3) $b=c+n/2$ and $a=n/2$. Then $I_n(a,b,c)+K_n(a,b,c)$ equals $1$ if at least one of these conditions is satisfied and $I_n(a,b,c)+K_n(a,b,c)$ equals $0$ otherwise. There are exactly three tuples $(a,b,c)$ that satisfy more than one of these conditions, namely $(0,0,0)$, $(n/2,n/2,0)$, and $(n/2,0,n/2)$.
\par
We now substitute~\eqref{eqn:IplusK} into~\eqref{eqn:F_from_L} and break the sum involving $I_n(a,b,c)+K_n(a,b,c)$ into six parts: three sums corresponding to the three cases and three sums to correct for the double counting of $(0,0,0)$, $(n/2,n/2,0)$, and $(n/2,0,n/2)$. Noting that the sums arising in cases (1) and (2) have the same value, as have the sums arising for the compensation of the double count of $(n/2,n/2,0)$ and $(n/2,0,n/2)$, we obtain
\[
\frac{1}{F(f_{r,t})}=-1+A+B+C-D_1-D_2-D_3+E,
\]
where
\begin{align*}
A=B&=\frac{1}{t^2n}\sums{0 \le j_1,j_2,j_3,j_4 < t \\ j_1+j_2=j_3+j_4}\;\sum_{b\in\mathbb{Z}/n\mathbb{Z}} \epsilon_b^{j_3-j_2}, \\
C&=\frac{1}{t^2n}\sums{0 \le j_1,j_2,j_3,j_4 < t \\ j_1+j_2=j_3+j_4} (-1)^{j_3-j_2}\sum_{c\in\mathbb{Z}/n\mathbb{Z}} \epsilon_c^{j_3+j_4+2 r}, \\
D_1&=\frac{1}{t^2n}\sums{0 \le j_1,j_2,j_3,j_4 < t \\ j_1+j_2=j_3+j_4} 1,\\
D_2=D_3&=\frac{1}{t^2n}\sums{0 \le j_1,j_2,j_3,j_4 < t \\ j_1+j_2=j_3+j_4} (-1)^{j_3-j_2},\\
E&=\frac{1}{t^2n}\sum_{a,b,c\,\in\,\mathbb{Z}/n\mathbb{Z}}M_n(a,b,c)\,\epsilon_{-a+b+c}^r\,\sums{0 \le j_1,j_2,j_3,j_4 < t \\ j_1+j_2=j_3+j_4}\epsilon_a^{-j_2}\epsilon_b^{j_3}\epsilon_c^{j_4}.
\end{align*}
As in the proof of~\cite[Theorem~4.2]{JedKatSch2013}, we have $-1+A+B-D_1+E\to 1/\varphi_0(0,T)$ if $t/n\to T$. Hence it remains to show that $C-D_2-D_3\to 0$ if $t/n\to T$.
\par
Since there are contributions to the first sum in $C$ only when $j_3+j_4=mn-2r$ for some $m\in\mathbb{Z}$, we obtain
\[
C=\frac{1}{t^2}\sum_{m\in\mathbb{Z}}\Bigg(\sum_{0\le j,mn-2r-j<t}(-1)^j\Bigg)^2.
\]
Therefore we have $\abs{C}\le 1/(tn)$ and so $C\to 0$ if $t/n\to T$. By writing $j_3=j_1+m$ for some $m\in\mathbb{Z}$ we find that
\[
D_2=\frac{1}{t^2n}\sum_{m\in\mathbb{Z}}\Bigg(\sum_{0\le j,j+m<t}(-1)^j\Bigg)^2.
\]
Hence $\abs{D_2}\le 1/(tn)$ and therefore $D_2+D_3\to0$ if $t/n\to T$. This completes the proof.
\end{proof}

\section{Some background on Gauss and Jacobi sums}

Let $\chi$ be a multiplicative character of $\mathbb{F}_q$. Throughout this paper, we use the common convention
\[
\chi(0)=\begin{cases}
0 & \text{if $\chi$ is nontrivial}\\
1 & \text{if $\chi$ is trivial}.
\end{cases}
\]
We define the \emph{canonical Gauss sum} of $\chi$ to be
\begin{equation}
G(\chi)=\sum_{y\in\mathbb{F}_q^*}\chi(y)\,e^{2\pi i\Tr_{q,p}(y)/p},   \label{eqn:def_gauss}
\end{equation}
where $p$ is the characteristic of $\mathbb{F}_q$ and $\Tr_{q,p}$ is the trace from $\mathbb{F}_q$ to $\mathbb{F}_p$. Below we summarise some basic facts about such Gauss sums (see~\cite[Chapter~5]{LidNie1997} or~\cite[Chapter~1]{BerEvaWil1998}, for example).
\begin{lemma}
Let $\chi$ be a multiplicative character of $\mathbb{F}_q$. Then the following hold.
\begin{enumerate}[(i)]
\label{lem:gauss}
\setlength{\itemsep}{1ex}
\item $G(\chi)=-1$ if $\chi$ is trivial.
\item $\abs{G(\chi)}=\sqrt{q}$ if $\chi$ is nontrivial.
\item $G(\chi)G(\overline{\chi})=\chi(-1)\,q$.
\end{enumerate}
\end{lemma}
\par
We also require the following deep result due to Katz~\cite[pp.~161--162]{Kat1988}.
\begin{lemma}
\label{lem:sum_of_gauss_sums}
Let $\alpha_1,\dots,\alpha_r,\beta_1,\dots,\beta_s$ be multiplicative characters of $\mathbb{F}_q$ such that $\alpha_1,\dots,\alpha_r$ do not arise by permuting $\beta_1,\dots,\beta_s$. Then
\begin{align*}
\Biggabs{
\sum_{\chi}G(\chi\alpha_1)\cdots\;
G(\chi\alpha_r)\overline{G(\chi\beta_1)}\cdots\;\overline{G(\chi\beta_s)}}\le \max(r,s)\,q^{(r+s+1)/2},
\end{align*}
where the sum runs over all multiplicative characters $\chi$ of $\mathbb{F}_q$.
\end{lemma}
\par
Now let $\psi$ and $\chi$ be multiplicative characters of $\mathbb{F}_q$. The \emph{Jacobi sum} corresponding to $\psi$ and $\chi$ is defined to be
\[
J(\psi,\chi)=\sum_{y\in\mathbb{F}_q}\psi(y)\chi(1-y).
\]
Below we summarise some basic facts about Jacobi sums (see~\cite[Chapter~5]{LidNie1997} or~\cite[Chapter~2]{BerEvaWil1998}, for example).
\begin{lemma}
\label{lem:jacobi}
Let $\psi$ and $\chi$ be multiplicative characters of $\mathbb{F}_q$. Then the following hold.
\begin{enumerate}[(i)]
\setlength{\itemsep}{1ex}
\item $J(\psi,\chi)=0$ if exactly one of $\psi$ or $\chi$ is trivial.
\item $\abs{J(\psi,\chi)}=1$ if $\psi$ and $\chi$ are nontrivial, but $\psi\chi$ is trivial.
\item $\abs{J(\psi,\chi)}=\sqrt{q}$ if all of $\psi$, $\chi$, and $\psi\chi$ are nontrivial.
\item $J(\psi,\chi)\,q=G(\psi)G(\chi)\overline{G(\psi\chi)}$ if $\psi$ and $\chi$ are nontrivial.
\item $J(\psi,\chi)J(\overline{\psi},\chi\psi)=\psi(-1)q$ if $\psi$ and $\chi$ are nontrivial.
\end{enumerate}
\end{lemma}

\section{Gordon-Mills-Welch difference sets}
\label{sec:gmw}

In this section we prove Theorem~\ref{thm:gmw}. If $D$ is a subset of a finite abelian group $G$ and $\chi$ is a character of $G$, we write
\[
\chi(D)=\sum_{d\in D}\chi(d).
\]
The following lemma is a standard (and easily verified).
\begin{lemma}
\label{lem:character_values_diff_set}
A $k$-subset $D$ of a finite abelian group $G$ of order $n$ is a difference set if and only if
\[
\abs{\chi(D)}^2=\frac{k(n-k)}{n-1}
\]
for all nontrivial characters $\chi$ of $G$.
\end{lemma}
\par
The following lemma gives the character values of Gordon-Mills-Welch difference sets and in particular gives an alternative proof of the main result of~\cite{GorMilWel1962} (although, for simplicity, we restrict $q$ to be even).
\begin{lemma}
\label{lem:character_values_gmw}
Let $q>2$ be a power of two and let $\mathbb{F}_s$ be a subfield of $\mathbb{F}_q$. Let~$A$ contain all elements $a\in\mathbb{F}_q$ with $\Tr_{q,s}(a)=1$, let $B$ be a subset of $\mathbb{F}_s^*$, and write $D=\{ab:a\in A,b\in B\}$. Let $\chi$ be a nontrivial character of $\mathbb{F}_q^*$ and let $\chi^*$ be its restriction to $\mathbb{F}_s^*$. Then
\[
\chi(D)=\begin{cases}
\dfrac{\chi^*(B)}{G(\chi^*)}G(\chi) & \text{for $\chi^*$ nontrivial}\\[3ex]
-\dfrac{\chi^*(B)}{s}G(\chi)        & \text{for $\chi^*$ trivial}.
\end{cases}
\]
In particular, if $B$ is a difference set in $\mathbb{F}_s^*$ and $\abs{B}=s/2$, then~$D$ is a difference set with parameters $(q-1,q/2,q/4)$.
\end{lemma}
\begin{proof}
We have
\[
\chi(D)=\sums{a\in\mathbb{F}_q\\\Tr_{q,s}(a)=1}\sum_{b\in B}\chi(ab)=E(\chi)\,\chi^*(B),
\]
where
\[
E(\chi)=\sums{a\in\mathbb{F}_q\\\Tr_{q,s}(a)=1}\chi(a)
\]
is an Eisenstein sum. It is known~\cite[pp.~391/400]{BerEvaWil1998} that
\[
E(\chi)=\begin{cases}
G(\chi)/G(\chi^*) & \text{for $\chi^*$ nontrivial}\\[1ex]
-G(\chi)/s        & \text{for $\chi^*$ trivial},
\end{cases}
\]
which proves the first statement of the lemma. The second statement follows from Lemmas~\ref{lem:character_values_diff_set} and~\ref{lem:gauss}.
\end{proof}
\par
We now prove Theorem~\ref{thm:gmw} by combining Theorem~\ref{thm:mf_from_L_1} with the following result.
\begin{proposition}
\label{pro:L_gmw}
Let $q>2$ be a power of two and let $f$ be a characteristic polynomial of a Gordon-Mills-Welch difference set in $\mathbb{F}_q^*$. Then
\[
\bigabs{L_f(a,b,c)-I_{q-1}(a,b,c)}\le\frac{2q^{5/2}}{(q-1)^3}
\]
for all $a,b,c\in\mathbb{Z}/(q-1)\mathbb{Z}$.
\end{proposition}
\begin{proof}
By definition, there exists a proper subfield $\mathbb{F}_s$ of $\mathbb{F}_q$ such that the underlying Gordon-Mills-Welch difference set is
\[
D=\{ab:a\in A,b\in B\},
\]
where $A$ contains all elements $a\in\mathbb{F}_q$ with $\Tr_{q,s}(a)=1$ and $B$ is a difference set in $\mathbb{F}_s^*$ with $\abs{B}=s/2$. Let~$\theta$ be a generator for $\mathbb{F}_q^*$ such that
\[
f(z)=\sum_{j=0}^{q-2}\mathbbm{1}_D(\theta^j)z^j.
\]
Let $\xi$ be the multiplicative character of $\mathbb{F}_q$ given by $\xi(\theta)=e^{2\pi i/(q-1)}$. It is readily verified that
\[
f(e^{2\pi ik/(q-1)})=\begin{cases}
1         & \text{for $k\equiv 0\pmod n$}\\[1ex]
2\xi^k(D) & \text{for $k\not\equiv 0\pmod n$}.
\end{cases}
\]
It then follows from Lemmas~\ref{lem:character_values_gmw} and~\ref{lem:gauss}~(i) that
\[
f(e^{2\pi ik/(q-1)})=C_kG(\xi^k),
\]
where $C_k$ has unit magnitude for all $k$ and depends only on $k$ modulo $s-1$. Therefore
\begin{equation}
L_f(a,b,c)=\frac{1}{(q-1)^3}\sum_{\chi}G(\chi)G(\chi\xi^a)\overline{G(\chi\xi^b)G(\chi\xi^c)}\;C_\chi(a,b,c),   \label{eqn:L_gauss}
\end{equation}
where the sum is over all multiplicative characters $\chi$ of $\mathbb{F}_q$ and $C_\chi(a,b,c)$ has unit magnitude. Using Lemma~\ref{lem:gauss}, we obtain
\[
L_f(a,b,c)=\begin{cases}
1+\frac{q-2}{(q-1)^2} & \text{for $a=b=c=0$}\\[1ex]
1-\frac{1}{(q-1)^2}   & \text{for $\{0,a\}=\{b,c\}$ and $a\ne 0$},
\end{cases}
\]
which proves the desired result in the case that $I_{q-1}(a,b,c)=1$.
\par
Now assume that $\{0,a\}\ne\{b,c\}$, so that $I_{q-1}(a,b,c)=0$. We need to show that
\begin{equation}
\abs{L_f(a,b,c)}\le \frac{2q^{5/2}}{(q-1)^3}.   \label{eqn:L_GMW_small}
\end{equation}
Let $H$ be the subgroup of index $s-1$ of the character group of $\mathbb{F}_q^*$ and note that $H$ is not the trivial group since, by assumption, $s<q$. Then $C_\chi(a,b,c)$ is constant when $\chi$ ranges over a coset of $H$. Since $C_\chi(a,b,c)$ has unit magnitude, we find from~\eqref{eqn:L_gauss} and the triangle inequality that
\begin{equation}
\abs{L_f(a,b,c)}\le\frac{s-1}{(q-1)^3}\,\max_{\phi}\,\Biggabs{\sum_{\chi\in H}\,G(\chi\phi)G(\chi\phi\xi^a)\overline{G(\chi\phi\xi^b)G(\chi\phi\xi^c)}},   \label{eqn:L_sum_H}
\end{equation}
where the maximum is over all multiplicative characters $\phi$ of $\mathbb{F}_q$. By the definition~\eqref{eqn:def_gauss} of a Gauss sum over $\mathbb{F}_q$, the inner sum can be written as
\[
\sum_{w,x,y,z\in \mathbb{F}_q^*}(-1)^{\Tr_{q,2}(w+x+y+z)}\,\xi^a(x)\overline{\xi^b(y)\xi^c(z)}\,\phi\Big(\frac{wx}{yz}\Big)\sum_{\chi\in H}\chi\Big(\frac{wx}{yz}\Big).
\]
For each $w,x,y,z\in\mathbb{F}_q^*$, we have
\[
\frac{s-1}{q-1}\sum_{\chi\in H}\chi\Big(\frac{wx}{yz}\Big)=\frac{1}{q-1}\sum_\chi\chi\Big(\frac{wx}{yz}\Big),
\]
where the sum on the right-hand side is over all multiplicative characters of~$\mathbb{F}_q$, since both sides equal either $0$ or $1$ depending on whether $wx=yz$ or not. Therefore, we can rewrite the inner sum of~\eqref{eqn:L_sum_H} as
\[
\frac{1}{s-1}\sum_\chi\,G(\chi\phi)G(\chi\phi\xi^a)\overline{G(\chi\phi\xi^b)G(\chi\phi\xi^c)},
\]
where $\chi$ now runs over all multiplicative characters of~$\mathbb{F}_q$. The magnitude of this expression is at most $\frac{2}{s-1}q^{5/2}$ by Lemma~\ref{lem:sum_of_gauss_sums}. Substitute into~\eqref{eqn:L_sum_H} to conclude that~\eqref{eqn:L_GMW_small} holds, as required.
\end{proof}

\section{Sidelnikov sets}
\label{sec:sidelnikov}

In this section we prove Theorem~\ref{thm:sidelnikov} by combining Theorem~\ref{thm:mf_from_L_2} with the following result.
\begin{proposition}
Let $q$ be an odd prime power and let $f$ be a characteristic polynomial of a Sidelnikov set in $\mathbb{F}_q^*$. Then
\[
\bigabs{L_f(a,b,c)-(I_{q-1}(a,b,c)+K_{q-1}(a,b,c))}\le \frac{23q^{5/2}}{(q-1)^3}
\]
for all $a,b,c\in\mathbb{Z}/(q-1)\mathbb{Z}$.
\end{proposition}
\begin{proof}
Let $\eta$ be the quadratic character of $\mathbb{F}_q$. By the definition of a Sidelnikov set~\eqref{eqn:def_sidelnikov}, there exists a generator~$\theta$ of $\mathbb{F}_q^*$ such that
\[
f(z)=z^{\frac{q-1}{2}}+\sum_{j=0}^{q-2}\eta(\theta^j+1)z^j,
\]
where, as usual, $\eta(0)=0$. Let $\xi$ be the multiplicative character of $\mathbb{F}_q$ given by $\xi(\theta)=e^{2\pi i/(q-1)}$. Then we have, for all $k\not\equiv 0\pmod{q-1}$,
\begin{align*}
f(e^{2\pi i k/(q-1)})&=(-1)^k+\sum_{j=0}^{q-2}\eta(\theta^j+1)\xi^k(\theta^j)\\
&=(-1)^k+\sum_{y\in\mathbb{F}_q}\eta(y+1)\xi^k(y)\\
&=(-1)^k+\xi^k(-1)\sum_{y\in\mathbb{F}_q}\eta(y)\xi^k(1-y)\\
&=(-1)^k(1+J(\eta,\xi^k)).
\end{align*}
On the other hand we have $f(1)=1-\eta(1)=0$. Therefore
\begin{equation}
L_f(a,b,c)=\frac{(-1)^{a+b+c}}{(q-1)^3}\sum_{\chi}J(\eta,\chi)J(\eta,\chi\xi^a)\overline{J(\eta,\chi\xi^b)J(\eta,\chi\xi^c)}+\Delta,   \label{eqn:L_sidelnikov}
\end{equation}
where the sum is over all multiplicative characters $\chi$ of $\mathbb{F}_q$ and $\abs{\Delta}\le 15q^{3/2}/(q-1)^2$, using Lemma~\ref{lem:jacobi}. If one of $b$ and $c$ equals $a$ and the other is zero, then by Lemma~\ref{lem:jacobi} the sum in~\eqref{eqn:L_sidelnikov} is between $(q-5)q^2$ and $(q-2)q^2$. If $a=(q-1)/2$ and $b=c+(q-1)/2$, then $\xi^a=\eta$ and $\xi^b=\xi^c\eta$ and by Lemma~\ref{lem:jacobi} (in particular (v)) the sum~\eqref{eqn:L_sidelnikov} is again at least $(q-5)q^2$ and at most $(q-2)q^2$. Since
\[
\frac{(q-5)q^2}{(q-1)^3}=1-\frac{2q^2+3q-1}{(q-1)^3},
\]
this establishes the cases in which either $I_{q-1}(a,b,c)$ or $K_{q-1}(a,b,c)$ equals~$1$.
\par
Now assume that $(a,b,c)$ is such that $I_{q-1}(a,b,c)$ and $K_{q-1}(a,b,c)$ are both zero. Equivalently, the multisets
\begin{equation}
\{\xi^0,\xi^a,\xi^b\eta,\xi^c\eta\}\quad\text{and}\quad\{\eta,\xi^a\eta,\xi^b,\xi^c\}   \label{eqn:sets_chars}
\end{equation}
are distinct. Use Lemmas~\ref{lem:jacobi} and~\ref{lem:gauss} to see that the sum in~\eqref{eqn:L_sidelnikov} equals
\begin{equation}
\frac{1}{q^2}\sums{\chi}G(\chi)G(\chi\xi^a)G(\chi\eta\xi^b)G(\chi\eta\xi^c)\overline{G(\chi\eta)G(\chi\eta\xi^a)G(\chi\xi^b)G(\chi\xi^c)}   \label{eqn:L_sidelnikov_gauss}
\end{equation}
plus an error term of magnitude at most $4q^{3/2}$, where the sum is over all multiplicative characters $\chi$ of $\mathbb{F}_q$. Since the multisets~\eqref{eqn:sets_chars} are distinct, we can apply Lemma~\ref{lem:sum_of_gauss_sums} to see that~\eqref{eqn:L_sidelnikov_gauss} is at most~$4q^{5/2}$. This shows that
\enlargethispage{4ex}
\[
\abs{L_f(a,b,c)}\le \frac{23q^{5/2}}{(q-1)^3},
\]
as required.
\end{proof}

\section{Cyclotomic constructions}
\label{sec:residues}

In this section we prove Theorem~\ref{thm:mf_cyclotomic} and Corollaries~\ref{cor:fourth_order} and~\ref{cor:sixth_order}. We shall use the following notation. Let $m$ be an even positive integer and let~$p$ be a prime satisfying $p\equiv 1\pmod m$. Let $\omega$ be a primitive element of $\mathbb{F}_p$ and let $C_0,C_1,\dots,C_{m-1}$ be the cyclotomic classes of $\mathbb{F}_p$ of order $m$ with respect to $\omega$. Let $S$ be an $m/2$-element subset of $\{0,1,\dots,m-1\}$ and let $D$ be the union of the $m/2$ cyclotomic classes $C_s$ with $s\in S$. Taking $1$ as a generator for the additive group of $\mathbb{F}_p$, a characteristic polynomial of $D$ is
\begin{equation}
f(z)=\sum_{j=0}^{p-1}\mathbbm{1}_D(j)z^j.   \label{eqn:f_with_g_1}
\end{equation}
The following lemma gives the evaluations of $f$ at $p$-th roots of unity.
\begin{lemma}
\label{lem:char_val_residue_poly}
Assume the notation as above and let $\chi$ be a multiplicative character of $\mathbb{F}_p$ of order $m$. Then
\[
f(e^{2\pi ik/p})=\frac{2}{m}\sum_{j=1}^{m-1}G(\chi^j)\,\overline{\chi^j(k)}\;\sum_{s\in S}\;\overline{\chi^j(\omega^s)}-1.
\]
\end{lemma}
\begin{proof}
Since $f(1)=\abs{D}-\abs{\mathbb{F}_p\setminus D}=-1$, the result holds for $k\equiv 0\pmod p$, so assume that $k\not\equiv 0\pmod p$. By definition we have
\begin{align*}
f(e^{2\pi ik/p})&=\sum_{y\in D}e^{2\pi iky/p}-\sum_{y\in\mathbb{F}_p\setminus D}e^{2\pi iky/p}\\
&=2\sum_{y\in D}e^{2\pi iky/p}-\sum_{y\in\mathbb{F}_p}e^{2\pi iky/p}\\
&=2\sum_{y\in D}e^{2\pi iky/p}\\
&=2\sum_{s\in S}\sum_{y\in C_s}e^{2\pi iky/p}.
\end{align*}
Writing $h=\frac{p-1}{m}$, the inner sum
can be written as
\begin{align*}
\sum_{y\in C_s}e^{2\pi iky/p}&=\sum_{j=0}^{h-1}e^{2\pi ik\omega^{mj+s}/p}\\
&=\frac{1}{m}\left(\sum_{y\in\mathbb{F}_p}e^{2\pi ik\,\omega^sy^m/p}-1\right).
\end{align*}
Since $\sum_{j=0}^{m-1}\chi^j(y)$ equals $m$ if $y$ is an $m$-th power and equals zero otherwise, we have, for each $a\in\mathbb{F}_p^*$,
\begin{align*}
\sum_{y\in\mathbb{F}_p}e^{2\pi iay^m/p}
&=\sum_{y\in\mathbb{F}_p}e^{2\pi iay/p}\,\sum_{j=0}^{m-1}\chi^j(y)\\
&=\sum_{j=0}^{m-1}\sum_{y\in\mathbb{F}_p}e^{2\pi iy/p}\,\chi^j(y)\,\overline{\chi^j(a)}.
\end{align*}
For $j=0$, the inner sum equals zero, so we can let the outer sum start with $j=1$. Then all involved  multiplicative characters are nontrivial and we can restrict the summation range of the inner sum to $\mathbb{F}_p^*$. Therefore
\[
\sum_{y\in\mathbb{F}_p}e^{2\pi iay^m/p}=\sum_{j=1}^{m-1}G(\chi^j)\,\overline{\chi^j(a)},
\]
which gives the desired result.
\end{proof}
\par
Our next result estimates $L_f$ for $f$ given in~\eqref{eqn:f_with_g_1} at all points, but $(0,0,0)$.
\begin{proposition}
\label{pro:L_cyclotomic}
With the notation as above, we have
\[
\bigabs{L_f(a,b,c)-(I_p(a,b,c)+\nu J_p(a,b,c))}\le 18(m-1)^4p^{-1/2}
\]
for all $a,b,c\in\mathbb{Z}/p\mathbb{Z}$ with $(a,b,c)\ne(0,0,0)$, where
\[
\nu=\begin{cases}
1                  & \text{for $\frac{p-1}{m}$ even}\\[1ex]
(\frac{4N}{m}-1)^2 & \text{for $\frac{p-1}{m}$ odd}
\end{cases}
\]
and
\[
N=\bigabs{\{(s,s')\in S\times S:s-s'=m/2\}}.
\]
\end{proposition}
\begin{proof}
Let $\chi$ be a multiplicative character of $\mathbb{F}_p$ of order $m$ and write
\[
K(\chi^j)=\frac{2}{m}\sum_{s\in S}\overline{\chi^j(\omega^s)}.
\]
From Lemma~\ref{lem:char_val_residue_poly} we find that
\begin{equation}
f(e^{2\pi ik/p})=\sum_{j=1}^{m-1}G(\chi^j)K(\chi^j)\overline{\chi^j(k)}-1.   \label{eqn:f_pth_root}
\end{equation}
Hence $L_f(a,b,c)$ equals
\begin{multline}
\frac{1}{p^3}\sum_{j_1,j_2,j_3,j_4=1}^{m-1}G(\chi^{j_1})G(\chi^{j_2})\overline{G(\chi^{j_3})G(\chi^{j_4})}\,K(\chi^{j_1})K(\chi^{j_2})\overline{K(\chi^{j_3})K(\chi^{j_4})}\\
\times\sum_{k\in\mathbb{F}_p}\overline{\chi^{j_1}(k)\chi^{j_2}(k+a)}\chi^{j_3}(k+b)\chi^{j_4}(k+c)+\Delta,   \label{eqn:L_residue_poly}
\end{multline}
where $\abs{\Delta}\le 15(m-1)^4p^{-1/2}$, using that the magnitude of the sum on the right-hand side of~\eqref{eqn:f_pth_root} is at most $(m-1)p^{1/2}$ by Lemma~\ref{lem:gauss}. First consider the case that $b=0$ and $c=a\ne 0$, so that $I_p(a,b,c)=1$ and $J_p(a,b,c)=0$. Then the inner sum in~\eqref{eqn:L_residue_poly} is
\[
\sum_{k\in\mathbb{F}_p}\chi^{j_3-j_1}(k)\chi^{j_4-j_2}(k+a).
\]
This sum either has magnitude at most $\sqrt{p}$ by the Weil bound (see~\cite[Theorem~5.41]{LidNie1997}, for example) or equals $p$. Since $a\ne 0$, the latter case occurs if and only if $j_1\equiv j_3\pmod m$ and $j_2\equiv j_4\pmod m$. Therefore $L_f(a,0,a)$ equals
\[
\frac{1}{p^2}\left(\sum_{j=1}^{m-1}\abs{G(\chi^j)}^2\,\abs{K(\chi^j)}^2\right)^2
\]
plus an error term of magnitude at most $16(m-1)^4p^{-1/2}$. Then we find from Lemma~\ref{lem:gauss} and
\[
\sum_{j=1}^{m-1}\bigabs{K(\chi^j)}^2=1
\]
that the desired result holds for $b=0$ and $c=a\ne 0$. The case $c=0$ and $b=a\ne 0$ is completely analogous.
\par
Now assume that $a=0$ and $c=b\ne 0$, so that $I_p(a,b,c)=0$ and $J_p(a,b,c)=1$. Then the inner sum in~\eqref{eqn:L_residue_poly} equals
\[
\sum_{k\in\mathbb{F}_p}\overline{\chi^{j_1+j_2}(k)}\chi^{j_3+j_4}(k+b).
\]
As before, this sum either has magnitude at most $\sqrt{p}$ or equals~$p$, where the latter case occurs if and only if $j_1\equiv -j_2\pmod m$ and $j_3\equiv -j_4\pmod m$. Hence $L_f(0,b,b)$ equals
\begin{equation}
\frac{1}{p^2}\Biggabs{\sum_{j=1}^{m-1}G(\chi^j)G(\overline{\chi^j})\,K(\chi^j)K(\overline{\chi^j})}^2   \label{eqn:L_residue_poly_aa}
\end{equation}
plus an error term of magnitude at most $16(m-1)^4p^{-1/2}$. From Lemma~\ref{lem:gauss} we find that~\eqref{eqn:L_residue_poly_aa} equals
\[
\Biggabs{\sum_{j=1}^{m-1}\chi^j(-1)\,\abs{K(\chi^j)}^2}^2=\Bigg(\sum_{j=1}^{m-1}(-1)^{\frac{j(p-1)}{m}}\Biggabs{\frac{2}{m}\,\sum_{s\in S} e^{2\pi ijs/m}}^2\Bigg)^2.
\]
A standard calculation then shows that this expression equals $\nu$. This proves the desired result in the case that $a=0$ and $c=b\ne 0$.
\par
Now assume that $0,a,b,c$ do not form two pairs of equal elements. In this case, we invoke the Weil bound again to conclude that the inner sum in~\eqref{eqn:L_residue_poly} is at most $3\sqrt{p}$ in magnitude. Therefore we have by Lemma~\ref{lem:gauss}
\[
\abs{L_f(a,b,c)}\le 18(m-1)^4p^{-1/2},
\]
which completes the proof.
\end{proof}
\par
\begin{proof}[Proof of Theorem~\ref{thm:mf_cyclotomic}]
Without loss of generality, we may choose $1$ as a generator for the additive group of $\mathbb{F}_p$ and take~\eqref{eqn:f_with_g_1} as a characteristic polynomial of $D$ (if the generator is $v$, then replace $D$ by $v^{-1}D$).
\par
We shall deduce Theorem~\ref{thm:mf_cyclotomic} from Theorem~\ref{thm:mf_from_L_1}. Proposition~\ref{pro:L_cyclotomic} takes care of all values of $L_f(a,b,c)$ in the condition of Theorem~\ref{thm:mf_from_L_1}, except when $(a,b,c)=(0,0,0)$. We shall show that our assumption~\eqref{eqn:cond_Ru} takes care of the latter case. Writing
\[
R_u=\sum_{y\in\mathbb{F}_p}\mathbbm{1}_D(y)\mathbbm{1}_D(y+u),
\]
a standard calculation gives
\[
\abs{f(e^{2\pi ik/p})}^2=\sum_{u\in\mathbb{F}_p}R_u\,e^{-2\pi iku/p}
\]
and therefore, by Parseval's identity,
\[
\frac{1}{p}\sum_{k\in\mathbb{F}_p}\abs{f(e^{2\pi ik/p})}^4=\sum_{u\in\mathbb{F}_p}R_u^2.
\]
A counting argument shows that
\[
R_u=4\,\abs{(D+u)\cap D}-(p-2).
\]
Therefore, since $2\abs{D}=p-1$, we find that  $L_f(0,0,0)$ equals
\[
\frac{1}{p^3}\sum_{k\in\mathbb{F}_p}\abs{f(e^{2\pi ik/p})}^4=
1+\frac{1}{p^2}\sum_{u\in\mathbb{F}_p^*}\big(4\,\abs{(D+u)\cap D}-(p-2)\big)^2.
\]
Now our assumption~\eqref{eqn:cond_Ru} together with Proposition~\ref{pro:L_cyclotomic} imply that the condition of Theorem~\ref{thm:mf_from_L_1} is satisfied, which proves Theorem~\ref{thm:mf_cyclotomic}.
\end{proof}
\par
Next we show how to deduce Corollaries~\ref{cor:fourth_order} and~\ref{cor:sixth_order} from Theorem~\ref{thm:mf_cyclotomic}. First consider Corollary~\ref{cor:fourth_order}. As explained in Section~\ref{sec:results}, we just need to consider the case that $D=C_0\cup C_1$. The cyclotomic numbers of order four have been already determined by Gauss and can be found, for example, in~\cite[Theorem 2.4.1]{BerEvaWil1998}. These numbers depend on the representation $p=x^2+4y^2$ and on the parity of $(p-1)/4$. They also depend on the choice of the primitive element in $\mathbb{F}_p$ used to define the cyclotomic classes, but this is encapsulated in the fact that $y$ is only unique up to sign. Using the cyclotomic numbers of order four, we obtain the numbers $\abs{(D+u)\cap D}$, shown in Table~\ref{tab:cyc_num_4}. For example, if $u\in C_1$, then $u^{-1}\in C_3$ and $\abs{(D+u)\cap D}$ equals
\[
\abs{(C_3+1)\cap C_3}+\abs{(C_3+1)\cap C_0}+\abs{(C_0+1)\cap C_3}+\abs{(C_0+1)\cap C_0}.
\]
From the data in Table~\ref{tab:cyc_num_4} we conclude that the assumption $y^2(\log p)^3/p\to 0$ as $p\to\infty$ in Corollary~\ref{cor:fourth_order} implies the condition~\eqref{eqn:cond_Ru} in Theorem~\ref{thm:mf_cyclotomic}. It is also readily verified that $\nu=1$, which proves Corollary~\ref{cor:fourth_order}.
\begin{table}
\caption{The numbers $4\abs{(D+u)\cap D}-(p-2)$ for $D=C_0\cup C_1$ and primes~$p$ of the form $p=x^2+4y^2$.}
\label{tab:cyc_num_4}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{c||c|c}
\hline
$D$         & $\frac{p-1}{4}$ even & $\frac{p-1}{4}$ odd\\ \hline\hline
$u\in C_0$  & $-3+2y$ & $-1-2y$\\
$u\in C_1$  & $-3-2y$ & $-1+2y$\\
$u \in C_2$ & $ 1+2y$ & $-1-2y$\\
$u \in C_3$ & $ 1-2y$ & $-1+2y$\\\hline
\end{tabular}
\end{table}
\par
Now consider Corollary~\ref{cor:sixth_order}. Then it suffices to consider the cases that $D$ is one of the following sets
\[
C_0\cup C_1\cup C_2,\quad C_0\cup C_1\cup C_3,\quad C_0\cup C_1\cup C_4.
\]
The cyclotomic numbers of order six have been determined by Dickson~\cite{Dic1935} (see also~\cite{Hal1956} for $(p-1)/6$ odd and~\cite{Whi1960} for $(p-1)/6$ even). These numbers depend on the representation of $p$ as a sum of a square and three times a square (every prime congruent to $1$ modulo~$3$ can be represented in this way) and on the cubic character of~$2$. Since~$p$ is of the form $x^2+27y^2$, we know~\cite[Theorem~2.6.4]{BerEvaWil1998} that $2$ is a cube in $\mathbb{F}_p$. In this case, the numbers $\abs{(D+u)\cap D}$ are given in Tables~\ref{tab:cyc_num_6_odd} and~\ref{tab:cyc_num_6_even} (again $y$ is only unique up to sign, corresponding to different primitive elements in~$\mathbb{F}_p$). We again conclude that the assumption $y^2(\log p)^3/p\to 0$ as $p\to\infty$ in Corollary~\ref{cor:sixth_order} implies the condition~\eqref{eqn:cond_Ru} in Theorem~\ref{thm:mf_cyclotomic}. The proof of Corollary~\ref{cor:sixth_order} is completed by checking that $\nu=1$ if $D$ is of Paley type and $\nu=1/9$ if $D$ is of
Hall type.
\begin{table}
\caption{The numbers $4\abs{(D+u)\cap D}-(p-2)$ for primes of the form $x^2+27y^2$ and $(p-1)/6$ odd.}
\label{tab:cyc_num_6_odd}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{c||c|c|c}
\hline
$D$        & $C_0\cup C_1\cup C_2$ & $C_0\cup C_1\cup C_3$ & $C_0\cup C_2\cup C_3$ \\\hline\hline
$u\in C_0$ & $-1+8y$ & $-3+2y$ & $-3-2y$\\
$u\in C_1$ & $-1$    & $-1$    & $1+2y$ \\
$u\in C_2$ & $-1-8y$ & $1-2y$  & $-1$   \\
$u\in C_3$ & $-1+8y$ & $-3+2y$ & $-3-2y$\\
$u\in C_4$ & $-1$    & $-1$    & $1+2y$ \\
$u\in C_5$ & $-1-8y$ & $1-2y$  & $-1$   \\\hline
\end{tabular}
\end{table}
\begin{table}
\caption{The numbers $4\abs{(D+u)\cap D}-(p-2)$ for primes of the form $x^2+27y^2$ and $(p-1)/6$ even.}
\label{tab:cyc_num_6_even}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{c||c|c|c}
\hline
$D$        & $C_0\cup C_1\cup C_2$ & $C_0\cup C_1\cup C_3$ & $C_0\cup C_2\cup C_3$ \\\hline\hline
$u\in C_0$ & $-3+8y$ & $-3+6y$ & $-3+2y$\\
$u\in C_1$ & $-3$    & $-3-4y$ & $1-2y$\\
$u\in C_2$ & $-3-8y$ & $1+2y$  & $-3+4y$\\
$u\in C_3$ & $1+8y$  & $-3-2y$ & $-3-6y$\\
$u\in C_4$ & $1$     & $1+4y$  & $1+6y$\\
$u\in C_5$ & $1-8y$  & $1-6y$  & $1-4y$\\\hline
\end{tabular}
\end{table}

\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}

redpajama_set_name: RedPajamaArXiv


text:
#pragma once

#include <aws/gamelift/server/model/GameSession.h>
#include <aws/gamelift/server/LogParameters.h>
#include <functional>

namespace Aws
{
namespace GameLift
{
namespace Server
{
#ifndef GAMELIFT_USE_STD
    typedef void(*StartGameSessionFn)(Aws::GameLift::Server::Model::GameSession, void*);
    typedef void(*ProcessTerminateFn)(void*);
    typedef bool(*HealthCheckFn)(void*);
#endif

    class ProcessParameters
    {
#ifdef GAMELIFT_USE_STD
    public:
        ProcessParameters() :
		m_onStartGameSession(nullptr),
		m_onProcessTerminate(nullptr),
		m_onHealthCheck(nullptr),
		m_port(-1),
		m_logParameters(LogParameters())
		{}

        ProcessParameters(const std::function<void(Aws::GameLift::Server::Model::GameSession)> onStartGameSession,
            const std::function<void()> onProcessTerminate,
            const std::function<bool()> onHealthCheck,
            int port,
            const Aws::GameLift::Server::LogParameters logParameters) : m_onStartGameSession(onStartGameSession),
            m_onProcessTerminate(onProcessTerminate),
            m_onHealthCheck(onHealthCheck),
            m_port(port),
            m_logParameters(logParameters){}

        AWS_GAMELIFT_API std::function<void(Aws::GameLift::Server::Model::GameSession)> getOnStartGameSession() const { return m_onStartGameSession; }
        AWS_GAMELIFT_API std::function<void()> getOnProcessTerminate() const { return m_onProcessTerminate; }
        AWS_GAMELIFT_API std::function<bool()> getOnHealthCheck() const { return m_onHealthCheck; }
        AWS_GAMELIFT_API int getPort() const { return m_port; }
        AWS_GAMELIFT_API Aws::GameLift::Server::LogParameters getLogParameters() const { return m_logParameters; }

    private:
        std::function<void(Aws::GameLift::Server::Model::GameSession)> m_onStartGameSession;
        std::function<void()> m_onProcessTerminate;
        std::function<bool()> m_onHealthCheck;
        int m_port;
        Aws::GameLift::Server::LogParameters m_logParameters;
#else
    public:
        ProcessParameters() :
		m_onStartGameSession(nullptr),
        m_onProcessTerminate(nullptr),
        m_onHealthCheck(nullptr),
        m_startGameSessionState(nullptr),
        m_processTerminateState(nullptr),
        m_healthCheckState(nullptr),
		m_port(-1),
		m_logParameters(LogParameters())
		{ }

        ProcessParameters(
            StartGameSessionFn onStartGameSession,
            void* startGameSessionState,
            ProcessTerminateFn onProcessTerminate,
            void* processTerminateState,
            HealthCheckFn onHealthCheck,
            void* healthCheckState,
            int port,
            const Aws::GameLift::Server::LogParameters logParameters) : m_onStartGameSession(onStartGameSession),
            m_startGameSessionState(startGameSessionState),
            m_onProcessTerminate(onProcessTerminate),
            m_processTerminateState(processTerminateState),
            m_onHealthCheck(onHealthCheck),
            m_healthCheckState(healthCheckState),
            m_port(port),
            m_logParameters(logParameters){}

        AWS_GAMELIFT_API StartGameSessionFn getOnStartGameSession() const { return m_onStartGameSession; }
        AWS_GAMELIFT_API void* getStartGameSessionState() const { return m_startGameSessionState; }
        AWS_GAMELIFT_API ProcessTerminateFn getOnProcessTerminate() const { return m_onProcessTerminate; }
        AWS_GAMELIFT_API void* getProcessTerminateState() const { return m_processTerminateState; }
        AWS_GAMELIFT_API HealthCheckFn getOnHealthCheck() const { return m_onHealthCheck; }
        AWS_GAMELIFT_API void* getHealthCheckState() const { return m_healthCheckState; }
        AWS_GAMELIFT_API int getPort() const { return m_port; }
        AWS_GAMELIFT_API Aws::GameLift::Server::LogParameters getLogParameters() const { return m_logParameters; }

    private:
        StartGameSessionFn m_onStartGameSession;
        ProcessTerminateFn m_onProcessTerminate;
        HealthCheckFn m_onHealthCheck;
        void* m_startGameSessionState;
        void* m_processTerminateState;
        void* m_healthCheckState;
        int m_port;
        Aws::GameLift::Server::LogParameters m_logParameters;
#endif
    };
} // namespace Server
} // namespace GameLift
} // namespace Aws

redpajama_set_name: RedPajamaGithub