Translation invariance in groups of prime order

TL;DR

This paper proves a bound on the number of shifts of a subset in a prime order group that minimally expand the set, extending previous integer results to prime groups with optimal bounds.

Contribution

It extends known results about translation invariance and minimal expansion from integers to groups of prime order, establishing optimal bounds and discussing potential relaxations.

Findings

Bound on the number of non-zero elements with small set difference after translation.

The bound is optimal up to a constant factor.

The result generalizes earlier integer-based theorems to prime order groups.

Abstract

We prove that there is an absolute constant $c>0$ with the following property: if $Z/pZ$ denotes the group of prime order $p$, and a subset $A\subset Z/pZ$ satisfies $1<|A|<p/2$, then for any positive integer $m<\min\{c|A|/\ln|A|,\sqrt{p/8}\}$ there are at most $2m$ non-zero elements $b\in Z/pZ$ with $|(A+b)\setminus A|\le m$. This (partially) extends onto prime-order groups the result, established earlier by S. Konyagin and the present author for the group of integers. We notice that if $A\subset Z/pZ$ is an arithmetic progression and $m<|A|<p/2$, then there are exactly $2m$ non-zero elements $b\in Z/pZ$ with $|(A+b)\setminus A|\le m$. Furthermore, the bound $c|A|/\ln|A|$ is best possible up to the value of the constant $c$. On the other hand, it is likely that the assumption $m<\sqrt{p/8}$ can be dropped or substantially relaxed.