Restricted sumsets in multiplicative subgroups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918459754938368 |
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| author | Yip, Chi Hoi |
| author_facet | Yip, Chi Hoi |
| contents | 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 $A \hat{+} A$, 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_10950 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Restricted sumsets in multiplicative subgroups Yip, Chi Hoi Number Theory Combinatorics Primary 11B30, 11P70, Secondary 11B13, 05C25 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 $A \hat{+} A$, 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 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. |
| title | Restricted sumsets in multiplicative subgroups |
| topic | Number Theory Combinatorics Primary 11B30, 11P70, Secondary 11B13, 05C25 |
| url | https://arxiv.org/abs/2309.10950 |