Arithmetic Structure in Sparse Difference Sets

Abstract

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 $\{v_1,...,v_{\ell }\}$ of vectors in ${\mathbb{Z}}^d$, we show that if $A\subset [1,N{]}^d$ with $|A|/N^d\ge C{N}^{-1/\ell }$, then there necessarily exists $r\ne 0$ such that $\{r{v}_1,...,r{v}_{\ell }\}\subseteq A-A$.

7 pages, to appear in the Journal of Number Theory.