On Kelly's lemma for infinite sets of integers

For a set of integers A subset of or equal to Z and k greater than or equal to 1 the k-deck of A is the function d(A,k) defined on sets S of k integers by d(A,k)(S) = \{i epsilon Z\{s + i\s epsilon S} subset of or equal to A}\. For k greater than or equal to 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 Kelly's Lemma for finite graphs. (C) 2002 Elsevier Science B.V. All rights reserved.