Arithmetic structure in sparse difference sets

Using a slight modification of an argument of Croot, Ruzsa and Schoen we establish a quantitative result on the existence of a dilated copy of any given configuration of integer points in sparse difference sets. More precisely, given any configuration {v1,…,vl} of vectors in Zd, we show that if A⊂[1,N]d with |A|/Nd⩾CN−1/l, then there necessarily exists r≠0 such that {rv1,…,rvl}⊆A−A.