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⊂d[1,N] with |A|/Nd⩾CN−1/ℓ, then there necessarily exists r≠0 such that {rv1,…,rvℓ}⊆A−A.