Sums of dilates in ${\mathbb{Z}}_p$

Abstract

We consider the problem of sums of dilates in groups of prime order. We show that given $A\subset \mathbb{Z}p$ of sufficiently small density then $$|{\lambda }_{1}A+{\lambda }_{2}A+...+{\lambda }_{k}A|\ge (\sum _i|{\lambda }_i|)|A|-o(|A|),$$ whereas on the other hand, for any $\epsilon >0$, we construct subsets of density $1/2-\epsilon$ such that $|A+\lambda A|\le (1-\delta )p$, showing that there is a very different behaviour for subsets of large density.