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

TL;DR

This paper investigates the behavior of sumsets of dilates in prime order groups, establishing lower bounds for small subsets and constructing examples for large subsets that demonstrate contrasting behaviors.

Contribution

It provides new bounds for sumsets of dilates in prime groups and constructs large subsets illustrating different sumset behaviors.

Findings

Lower bounds for small subsets' sumsets of dilates

Existence of large subsets with small sumsets

Contrasting behaviors based on subset density

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.