We establish the restricted sumset analog 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 Fqβ cannot be written as a
restricted sumset A+^βA, 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 analog of van
Lint-MacWilliams' conjecture for restricted sumsets, equivalently, an analog of
Erd\H{o}s-Ko-Rado theorem in a family of Cayley sum graphs.Comment: 23 page