On Kellyʹs lemma for infinite sets of integers

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 bydA,k(S)=|{i∈Z | {s+i | s∈S}⊆A}|.For k≥3 we prove a sufficient condition implying that two sets with the same k-deck also have the same (k−1)-deck. This is an analogue of Kellyʹs Lemma for finite graphs.