## unique sums in a finite direct product of sets of integers

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I am an algebraist, and I am wondering if there is a definition for the following:

Let $A_1$, $A_2$, $\ldots, A_n$ be sets of integers (or more generally, subsets of a group $G$). Say that (for the purposes of this question) $A_1\times A_2\cdots\times A_n$ is special provided that whenever $a_1+a_2+\cdots+a_n=b_1+b_2\cdots+b_n$ with each $a_i,b_i\in A_i$, $1\leq i\leq n$, then $a_i=b_i$ for all $i$, $1\leq i\leq n$.

Is there some terminology for this property? Any information would be appreciated.

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In the case that $A_1,\dotsc,A_n$ are singletons, this notion as known as unique-sum sets (see e.g. cmc11.uni-jena.de/proceedings/frisco.pdf). – Martin Brandenburg Dec 20 at 8:07
@Martin: I suspect there must be something wrong in your statement because if $A_1, \ldots, A_n$ are singletons, then also $A_1 \times \ldots \times A_n$ is a singleton, and hence the condition in the question is trivially true. – boumol Dec 20 at 8:26
I think Martin is referrring to $B_n[1]$ sets which would -- I think! -- be the case $A_1=A_2=\dots= A_n$ – Yemon Choi Dec 20 at 9:37
While what Yemon Choi says is also a pertinent related notion Martin Brandenburg meant something else. Namely, two elements sets (not singletons). The point is that one studies the question when a set (or also multiset) has the property that all the sums of elements of distinct subsets are actually distinct. This corresponds directly to the case that the A_i are of the form {0,a_i} in the present problem. Yet since the present problem is invarinat under shifts this also applies in case each A_i contains (at most) two elements. – quid Dec 20 at 11:30

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A more natural way to state the condition is that $|A_1+\cdots+A_h| = |A_1| \cdots |A_h|$ (the parameter $h$ is more common than $n$ here). These come up in additive combinatorics, but I don't know a name for them. If $A_1= \dots =A_h$, and you don't care about the ordering of the sum (i.e., one can reorder the $b_i$ so that $a_i=b_i$), these are called "Sidon Sets", also $B_h$-sets. – Kevin O'Bryant Dec 20 at 19:30

## 1 Answer

There is a lot of interesting work using the term "Tiling" (sometimes " Algebraic Tiling"), especially when there are two factors although all the factors but one can be combined (I am thinking mainly of Abelian Groups). If $A$ is any infinite set of integers which increases quickly enough then there is a $B$ with $A \oplus B= \mathbb{Z}.$ For $A \oplus B=\mathbb{Z}$ with $A$ finite, there are results and open questions, cyclotomic polynomials are a useful tool For $A \oplus B= \mathbb{N}$ (bases for the positive integers) both $A$ and $B$ are highly structured. Evidently there are connections to Musical Canons.

There have uses of non-abelian factorizations for cryptography.

There are better and other references but that gives a few entry points

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