Sums of dilates in groups of prime order

TL;DR

This paper provides the first non-trivial estimate for the sum of dilates problem in groups of prime order, establishing lower bounds for the size of sumsets involving dilates under certain conditions.

Contribution

It introduces explicit bounds for sumsets of the form | extstyle A + t extstyle imes A| in prime order groups, extending previous results with new quantitative estimates.

Findings

For t ≠ 0, ±1, the sumset size exceeds (2 + θ_t)|A| minus a constant w(t).

Explicit example: for |t|=2, θ_2=0.08.

The bounds depend only on t and are valid for sufficiently small A relative to p.

Abstract

We obtain a first non-trivial estimate for the sum of dilates problem in the case of groups of prime order, by showing that if $t$ is an integer different from $0, 1$ or -1 and if $\A \subset \Zp$ is not too large (with respect to $p$), then $|\A+t\cdot \A|>(2+ \vartheta_t)|\A|-w(t)$ for some constant $w(t)$ depending only on $t$ and for some explicit real number $\vartheta_t >0$ (except in the case $|t|=3$). In the important case $|t|=2$, we may for instance take $\vartheta_2=0.08$.