## Return to Answer

2 added 287 characters in body

No, there does not exist such a $V$.

Let $W = V \cup (V \oplus r)$, and suppose $W = F \Delta Q$. Note that for $s \in {\mathbb Q}$, $W \cap (W \oplus s)$ is nonempty if and only if $s$ or $s-r$ or $s+r$ is an integer. But if $F$ contained an interval of positive length, $W \cap (W \oplus s)$ would be nonempty for all sufficiently small $|s|$. Thus $F$ is nowhere dense, and $W$ is of first category.
But this is impossible, because $[0,1] = \bigcup_{r \in {\mathbb Q}} (W \oplus r)$.

For your second question, the properties of Vitali sets (and finite unions of their translates) that this uses are:

1) there is a dense set of $r$ such that $V \cap (V \oplus r) = \emptyset$.

2) there is a countable set $S$ such that $\bigcup_{r \in S} (V \oplus r) = [0,1)$.

1

No, there does not exist such a $V$.

Let $W = V \cup (V \oplus r)$, and suppose $W = F \Delta Q$. Note that for $s \in {\mathbb Q}$, $W \cap (W \oplus s)$ is nonempty if and only if $s$ or $s-r$ or $s+r$ is an integer. But if $F$ contained an interval of positive length, $W \cap (W \oplus s)$ would be nonempty for all sufficiently small $|s|$. Thus $F$ is nowhere dense, and $W$ is of first category.
But this is impossible, because $[0,1] = \bigcup_{r \in {\mathbb Q}} (W \oplus r)$.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 32, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9136302471160889, "perplexity_flag": "head"}