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We establish the restricted sumset analog 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 
$A \hat {+} A$, 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 Erdős and Moser. We also prove an analog 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.
