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 $A\subset \Z{p}$ of sufficiently small density then $$\big| \lambda_{1}A+\lambda_{2}A+...+ \lambda_{k}A \big| \,\ge\,\bigg(\sum_{i}|\lambda_{i}|\bigg)|A|- o(|A|),$$ whereas on the other hand, for any $\epsilon>0$, we construct subsets of density $1/2-\epsilon$ such that $|A+\lambda A|\leq (1-\delta)p$, showing that there is a very different behaviour for subsets of large density.