An improved non-linear Roth-type theorem in finite fields
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866910179731177472 |
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| author | Lewko, Mark |
| author_facet | Lewko, Mark |
| contents | Let $F$ be a finite field of odd characteristic. We prove that any set $A\subset F$ with $|A|\geq C|F|^{5/6}$ contains a nontrivial quadratic progression $(x, x+y, x+y^2), y\neq 0.$ For prime fields, this improves the previous best-known exponent of $7/8$, due to Kavrut and Wu. Unlike some of the previous papers, which rely on Katz's deep multivariate exponential-sum estimates, our argument uses only one-variable Weil-type estimates. We also construct, over certain non-prime finite fields, progression-free sets of size $c|F|^{2/3}$. A key idea in the proof was suggested to the author by ChatGPT 5.5. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_27501 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An improved non-linear Roth-type theorem in finite fields Lewko, Mark Number Theory Combinatorics Let $F$ be a finite field of odd characteristic. We prove that any set $A\subset F$ with $|A|\geq C|F|^{5/6}$ contains a nontrivial quadratic progression $(x, x+y, x+y^2), y\neq 0.$ For prime fields, this improves the previous best-known exponent of $7/8$, due to Kavrut and Wu. Unlike some of the previous papers, which rely on Katz's deep multivariate exponential-sum estimates, our argument uses only one-variable Weil-type estimates. We also construct, over certain non-prime finite fields, progression-free sets of size $c|F|^{2/3}$. A key idea in the proof was suggested to the author by ChatGPT 5.5. |
| title | An improved non-linear Roth-type theorem in finite fields |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2604.27501 |