Restricted sumsets in multiplicative subgroups

Abstract

We establish the restricted sumset analogue of the celebrated conjecture of Sárközy 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 $A\widehat{+}A$, 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 Erdős and Moser. We also prove an analogue of van Lint-MacWilliams' conjecture for restricted sumsets, which appears to be the first analogue of Erdős-Ko-Rado theorem in a family of Cayley sum graphs.

23 pages,revised based on referee comments

References in corpus (3)

• Van Lint-MacWilliams' conjecture and maximum cliques in Cayley graphs over finite fields
• Quadratic residues and difference sets
• Additive decompositions of large multiplicative subgroups in finite fields

Cited by in corpus (2)

• Multiplicative structure of shifted multiplicative subgroups and its applications to Diophantine tuples
• Multiplicative irreducibility of small perturbations of the set of shifted $k$-th powers