Improved Bounds for the Freiman-Ruzsa Theorem
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915820854050816 |
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| author | Raghavan, Rushil |
| author_facet | Raghavan, Rushil |
| contents | Let $A$ be a finite subset of an abelian group $G$, and suppose that $|A+A|\leq K|A|$. We show that for any $ε>0$, there exists a constant $C_ε$ such that $A$ can be covered by at most $\exp(C_ε\log(2K)^{1+ε})$ translates of a convex coset progression with dimension at most $C_ε\log(2K)^{1+ε}$ and size at most $\exp(C_ε\log(2K)^{1+ε})|A|$. This falls just short of the Polynomial Freiman-Ruzsa conjecture, which asserts that this statement is true for $ε=0$, and improves on results of Sanders and Konyagin, who showed that this statement is true for all $ε>2$. To prove this result, we use a mixture of entropy methods and Fourier analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_11217 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improved Bounds for the Freiman-Ruzsa Theorem Raghavan, Rushil Number Theory Combinatorics Let $A$ be a finite subset of an abelian group $G$, and suppose that $|A+A|\leq K|A|$. We show that for any $ε>0$, there exists a constant $C_ε$ such that $A$ can be covered by at most $\exp(C_ε\log(2K)^{1+ε})$ translates of a convex coset progression with dimension at most $C_ε\log(2K)^{1+ε}$ and size at most $\exp(C_ε\log(2K)^{1+ε})|A|$. This falls just short of the Polynomial Freiman-Ruzsa conjecture, which asserts that this statement is true for $ε=0$, and improves on results of Sanders and Konyagin, who showed that this statement is true for all $ε>2$. To prove this result, we use a mixture of entropy methods and Fourier analysis. |
| title | Improved Bounds for the Freiman-Ruzsa Theorem |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2512.11217 |