The sum-product phenomenon for dense subsets of finite fields
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866915943728283648 |
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| author | Shao, Xuancheng |
| author_facet | Shao, Xuancheng |
| contents | Let $\mathbb{F}_p$ be a finite field of prime order $p$ and let $A \subset \mathbb{F}_p$ be a subset. In the dense regime when $|A| \geq αp$ for some $α\in (0,1)$, we determine the optimal constant $f(α)$ in the inequality $$ \max(|A+A|, |A\cdot A|) \geq (f(α) - o(1))p. $$ The proof relies on a structural result for sumsets of dense subsets, established via a regularity lemma in general finite abelian groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_17117 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The sum-product phenomenon for dense subsets of finite fields Shao, Xuancheng Number Theory Let $\mathbb{F}_p$ be a finite field of prime order $p$ and let $A \subset \mathbb{F}_p$ be a subset. In the dense regime when $|A| \geq αp$ for some $α\in (0,1)$, we determine the optimal constant $f(α)$ in the inequality $$ \max(|A+A|, |A\cdot A|) \geq (f(α) - o(1))p. $$ The proof relies on a structural result for sumsets of dense subsets, established via a regularity lemma in general finite abelian groups. |
| title | The sum-product phenomenon for dense subsets of finite fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2604.17117 |