Reconstructing Infinite Sets of Integers

Abstract For a set of integers A⊆Z and k⩾1 the k-deck of A is the function dA,k defined on sets S of k integers by d A,k (S)=∣{I∈ Z ∣ {s+I∣s∈S}⊆A}∣ . Our main result is that for k⩾3, a set for which the k-deck only takes finite values is determined up to translation by its k-deck and one finite non-zero value of its (k−1)-deck. This generalizes a result of Radcliffe and Scott (Electron. J. Combin.6 (1999), R20) which proved a weaker form of this statement for k=3. In order to establish this result, we generalize Kelly's Lemma for finite graphs to infinite sets of integers.