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PreprintCC BY 4.0arXiv

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 k2 determine at most 3(mn)2/3+m+n+2mn/p incidences, the last term being omitted in characteristic zero. Over the prime field 𝔽p the coefficient of mn/p can be replaced by $1$. The proof uses the polynomial method, and for m=n the bound is sharp up to an absolute constant over prime fields. As applications, over prime fields in which −1 is not a square we obtain the L2→ Lr extension estimate for the paraboloid in 𝔽p3 for r>10/3. Over every odd prime field, we show that a two-source extractor construction of Bourgain has exponentially small error at every min-entropy rate greater than 1/3. We also improve sum-product estimates for small sets in positive characteristic and obtain projection and Furstenberg estimates over prime fields. The incidence inequalities with exact constants have been formalized in Lean.

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Cite

BibTeXRIS
BibTeX
@misc{lewko2026szemeréditrotter,
  title = {{A Szemerédi-Trotter Theorem in Arbitrary Fields}},
  author = {Mark Lewko},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.27023},
  eprint = {2609.27023},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27023},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Szemerédi-Trotter Theorem in Arbitrary Fields
AU  - Mark Lewko
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.27023
UR  - https://arxiv.org/abs/2609.27023
AB  - Let k be a field of characteristic p≥0. We prove that m points and n lines in k^2 determine at most 3(mn)^2/3+m+n+2mn/p incidences, the last term being omitted in characteristic zero. Over the prime field 𝔽_p the coefficient of mn/p can be replaced by $1$. The proof uses the polynomial method, and for m=n the bound is sharp up to an absolute constant over prime fields. As applications, over prime fields in which −1 is not a square we obtain the L^2→ L^r extension estimate for the paraboloid in 𝔽_p^3 for r>10/3. Over every odd prime field, we show that a two-source extractor construction of Bourgain has exponentially small error at every min-entropy rate greater than 1/3. We also improve sum-product estimates for small sets in positive characteristic and obtain projection and Furstenberg estimates over prime fields. The incidence inequalities with exact constants have been formalized in Lean.
ER  -