Restricted sumsets in multiplicative subgroups

TL;DR

This paper proves new results on restricted sumsets in finite fields, including a sumset analogue of Sárkőzy's conjecture for nonzero squares and extensions to multiplicative subgroups and perfect powers.

Contribution

It establishes the first analogue of Erdős-Ko-Rado theorem for restricted sumsets in Cayley sum graphs and extends sumset results to various algebraic structures.

Findings

Nonzero squares in finite fields cannot be expressed as restricted sumsets for large primes.

Extended sumset results to multiplicative subgroups and perfect powers.

Proved an analogue of van Lint-MacWilliams' conjecture for restricted sumsets.

Abstract

We establish the restricted sumset analogue of the celebrated conjecture of S\'{a}rk\"{o}zy on additive decompositions of the set of nonzero squares over a finite field. More precisely, we show that if $q>13$ is an odd prime power, then the set of nonzero squares in $\mathbb{F}_q$ cannot be written as a restricted sumset $A \hat{+} A$, extending a result of Shkredov. More generally, we study restricted sumsets in multiplicative subgroups over finite fields as well as restricted sumsets in perfect powers (over integers) motivated by a question of Erd\H{o}s and Moser. We also prove an analogue of van Lint-MacWilliams' conjecture for restricted sumsets, which appears to be the first analogue of Erd\H{o}s-Ko-Rado theorem in a family of Cayley sum graphs.