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 ∈ 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 Zn 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 Zn 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 ‹√ n).