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Restricted sumsets in multiplicative subgroups

Part of: Additive number theory; partitions Sequences and sets Graph theory

Published online by Cambridge University Press:  09 January 2025

Chi Hoi Yip Open the ORCID record for Chi Hoi Yip [Opens in a new window]
Chi Hoi Yip*
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, United States
*
e-mail: cyip30@gatech.edu
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Abstract

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.

Keywords

Restricted sumsetadditive decompositionmultiplicative subgroupSidon setCayley sum graph

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity 11P70: Inverse problems of additive number theory, including sumsets
Secondary: 11B13: Additive bases, including sumsets 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 25
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The research of the author was supported in part by an NSERC fellowship.

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