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

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 \( \mathbb{Z}_n \) is sufficient to determine the set up to translation. This completes the work begun by Grünbaum and Moore [GM] as far as the \( 3 \)-deck is concerned. We additionally estimate from above the probability that for a random subset of \( \mathbb{Z}_n \), 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 \( O(1/\sqrt{n}) \).