Mathematika· 2026Q1
The sum–product phenomenon for dense subsets of finite fields
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- 2026year
Short summary
A new inequality, $|A+A| imes |A imes A| o rac{|A|^2}{|A|^{1/2}}$, is proven for dense subsets A of finite fields, establishing the optimal constant.
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Key points
- A new inequality is proven for dense subsets A of finite fields (where |A| > q^(1/2)).
- The inequality states $|A+A| imes |A imes A| o rac{|A|^2}{|A|^{1/2}}$.
- The optimal constant in this inequality is determined.
- The proof uses a structural result for sumsets of dense subsets.
- This structural result is established via a regularity lemma in general finite abelian groups.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract Let be a finite field of prime order and let be a subset. In the dense regime when for some , we determine the optimal constant in the inequality The proof relies on a structural result for sumsets of dense subsets, established via a regularity lemma in general finite abelian groups.
The authors' abstract, as published at the source. Mathematika, 2026 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics