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arXiv · 2309.10950

Restricted sumsets in multiplicative subgroups

Chi Hoi Yip
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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.

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Chi Hoi Yip. 2023-09-19. Restricted sumsets in multiplicative subgroups. https://doi.org/10.4153/s0008414x24000920

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