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On the determination of sets by their triple correlation in finite cyclic groups

Tamas Keleti·Mihail N. Kolountzakis
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Abstract

Let $G$ be a finite abelian group and $E$ a subset of it. Suppose that we know for all subsets $T$ of $G$ of size up to $k$ for how many $x \in G$ the translate $x+T$ is contained in $E$. This information is collectively called the $k$-deck of $E$. One can naturally extend the domain of definition of the $k$-deck to include functions on $G$. Given the group $G$ when is the $k$-deck of a set in $G$ sufficient to determine the set up to translation? The 2-deck is not sufficient (even when we allow for reflection of the set, which does not change the 2-deck) and the first interesting case is $k=3$. We further restrict $G$ to be cyclic and determine the values of $n$ for which the 3-deck of a subset of $\ZZ_n$ is sufficient to determine the set up to translation. This completes the work begun by Grünbaum and Moore as far as the 3-deck is concerned. We additionally estimate from above the probability that for a random subset of $\ZZ_n$ there exists another subset, not a translate of the first, with the same 3-deck. We give an exponentially small upper bound when the previously known one was $O(1\bigl / \sqrt{n})$.

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Tamas Keleti, Mihail N. Kolountzakis. 2006-03-16. On the determination of sets by their triple correlation in finite cyclic groups. https://arxiv.org/abs/math/0603415

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