Generalised Fermat equations in dense variables over finite fields and rings
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908743793377280 |
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| author | Chow, Sam Lim, Zi Li Mudgal, Akshat |
| author_facet | Chow, Sam Lim, Zi Li Mudgal, Akshat |
| contents | Let $A$ be a sufficiently dense subset of a finite field $\mathbb F_q$ or a finite, cyclic ring $\mathbb Z/ N\mathbb Z$. Assuming that $q$ and $N$ have no small prime divisors, we show that generalised Fermat equations have the expected number of solutions over $A$. We further show that our density threshold is optimal. Our proofs involve average Fourier decay for Bohr sets, mixed character sum bounds, equidistribution of polynomial sequences, popular Cauchy--Davenport lemmas, and a regularity-type lemma due to Semchankau. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_00135 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalised Fermat equations in dense variables over finite fields and rings Chow, Sam Lim, Zi Li Mudgal, Akshat Number Theory 11B30 (primary), 11D41, 11D45, 11P05, 11T23 (secondary) Let $A$ be a sufficiently dense subset of a finite field $\mathbb F_q$ or a finite, cyclic ring $\mathbb Z/ N\mathbb Z$. Assuming that $q$ and $N$ have no small prime divisors, we show that generalised Fermat equations have the expected number of solutions over $A$. We further show that our density threshold is optimal. Our proofs involve average Fourier decay for Bohr sets, mixed character sum bounds, equidistribution of polynomial sequences, popular Cauchy--Davenport lemmas, and a regularity-type lemma due to Semchankau. |
| title | Generalised Fermat equations in dense variables over finite fields and rings |
| topic | Number Theory 11B30 (primary), 11D41, 11D45, 11P05, 11T23 (secondary) |
| url | https://arxiv.org/abs/2601.00135 |