Almost Affine Invariance Over Prime Fields: Green Problem 90
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917491214647296 |
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| author | Ma, Jie Tang, Quanyu Xu, Max Wenqiang |
| author_facet | Ma, Jie Tang, Quanyu Xu, Max Wenqiang |
| contents | Let $A\subset \mathbb{F}_p$ with density 1/2. We call a set $A$ almost affine invariant under an affine transformation $ϕ(x)=ax+b$ if \[|A \triangle ϕ(A)| =o(p).\] We determine that, the threshold value of $K$ such that $A$ is almost affine invariant simultaneously under all $ϕ(x)$ with $|a|, |b|\le K$ and $a\neq 0$, is $K=o(\log p)$. This solves Ben Green's Open Problem 90. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13454 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Almost Affine Invariance Over Prime Fields: Green Problem 90 Ma, Jie Tang, Quanyu Xu, Max Wenqiang Combinatorics Dynamical Systems Group Theory Number Theory Let $A\subset \mathbb{F}_p$ with density 1/2. We call a set $A$ almost affine invariant under an affine transformation $ϕ(x)=ax+b$ if \[|A \triangle ϕ(A)| =o(p).\] We determine that, the threshold value of $K$ such that $A$ is almost affine invariant simultaneously under all $ϕ(x)$ with $|a|, |b|\le K$ and $a\neq 0$, is $K=o(\log p)$. This solves Ben Green's Open Problem 90. |
| title | Almost Affine Invariance Over Prime Fields: Green Problem 90 |
| topic | Combinatorics Dynamical Systems Group Theory Number Theory |
| url | https://arxiv.org/abs/2605.13454 |