Restricted sumsets in multiplicative subgroups

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.