The Polynomial Freiman-Ruzsa (Marton) Conjecture in Integers and Finite Fields via Spectral Stability
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arXiv
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| Formato: | Preprint |
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2025
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| _version_ | 1866908701312417792 |
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| author | Moghadam, Mohammad Taha Kazemi |
| author_facet | Moghadam, Mohammad Taha Kazemi |
| contents | We settle the Polynomial Freiman--Ruzsa (PFR/Marton) conjecture for the integers and for cyclic groups. More precisely, we show that if $A$ is a finite subset of $\mathbb{Z}$ or $\mathbb{Z}/N\mathbb{Z}$ with $|A+A| \le K|A|$, then there is a subgroup $H$ of index at most $K^{O(1)}$ such that $A$ is contained in at most $K^{O(1)}$ cosets of $H$. The proof is based on a new spectral stability dichotomy for the $L^4$ Fourier mass of $\mathbf{1}_A$: either this mass is concentrated on a span of size $K^{O(1)}$, or, after passing to a quotient of codimension $K^{O(1)}$, the doubling constant of the image of $A$ decreases by a definite power of $K$. Using Freiman modeling we transfer this dichotomy to cyclic groups, obtain polynomial Bogolyubov-type bounds, and deduce Marton's conjecture in $\mathbb{Z}$ and $\mathbb{Z}/N\mathbb{Z}$. As a corollary, we also recover and extend the finite-field formulation of Marton's conjecture: in odd characteristic we obtain a direct spectral proof, and together with the characteristic-2 result of Green, Gowers, Manners, and Tao this yields a complete resolution of the conjecture for all finite fields. For context beyond finite fields, we recall their theorem for abelian groups of bounded exponent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04433 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Polynomial Freiman-Ruzsa (Marton) Conjecture in Integers and Finite Fields via Spectral Stability Moghadam, Mohammad Taha Kazemi Combinatorics Number Theory 11B13, 11B30, 05D10, 05D99 We settle the Polynomial Freiman--Ruzsa (PFR/Marton) conjecture for the integers and for cyclic groups. More precisely, we show that if $A$ is a finite subset of $\mathbb{Z}$ or $\mathbb{Z}/N\mathbb{Z}$ with $|A+A| \le K|A|$, then there is a subgroup $H$ of index at most $K^{O(1)}$ such that $A$ is contained in at most $K^{O(1)}$ cosets of $H$. The proof is based on a new spectral stability dichotomy for the $L^4$ Fourier mass of $\mathbf{1}_A$: either this mass is concentrated on a span of size $K^{O(1)}$, or, after passing to a quotient of codimension $K^{O(1)}$, the doubling constant of the image of $A$ decreases by a definite power of $K$. Using Freiman modeling we transfer this dichotomy to cyclic groups, obtain polynomial Bogolyubov-type bounds, and deduce Marton's conjecture in $\mathbb{Z}$ and $\mathbb{Z}/N\mathbb{Z}$. As a corollary, we also recover and extend the finite-field formulation of Marton's conjecture: in odd characteristic we obtain a direct spectral proof, and together with the characteristic-2 result of Green, Gowers, Manners, and Tao this yields a complete resolution of the conjecture for all finite fields. For context beyond finite fields, we recall their theorem for abelian groups of bounded exponent. |
| title | The Polynomial Freiman-Ruzsa (Marton) Conjecture in Integers and Finite Fields via Spectral Stability |
| topic | Combinatorics Number Theory 11B13, 11B30, 05D10, 05D99 |
| url | https://arxiv.org/abs/2512.04433 |