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arXiv · 2609.27023

A Szemerédi-Trotter Theorem in Arbitrary Fields

Mark Lewko
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Abstract

Let $k$ be a field of characteristic $p>0$. We prove that $m$ points and $n$ lines in $k^2$ determine $O((mn)^{2/3}+m+n+mn/p)$ incidences. In characteristic zero the last term is omitted. The proof uses the polynomial method, and for $m=n$ the bound is sharp over prime fields. As applications, over prime fields with $p\equiv 3\pmod4$ we obtain the $L^2\to L^r$ extension estimate for the paraboloid in $F_p^3$ for $r>10/3$, and we show that Bourgain's paraboloid extractor extracts from independent sources of any min-entropy rate greater than $3/8=0.375$ with exponentially small error. We also improve sum-product estimates for small sets in positive characteristic.

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Mark Lewko. 2026-09-22. A Szemerédi-Trotter Theorem in Arbitrary Fields. https://arxiv.org/abs/2609.27023

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