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The Szemerédi-Trotter Estimate in Finite Field with its Applications
Abstract
We prove a sharp Szemerédi-Trotter estimate ℐ(A,ℒ)≲ |A||ℒ|/p+|A|2/3|ℒ|2/3+|A|+|ℒ| for prime finite field 𝔽=𝔽p by a new polynomial decomposition theorem. As applications, we first prove the sharp Furstenberg set estimate in 𝔽2. Secondly, we improve sum-product estimate max{|A+A|,|A· A|}≳min{(p|A|)1/2,|A|5/4}, A⊂𝔽. Finally, we improve the Fourier restriction estimate R*(2→α) holds for α>10/3 in 𝔽3 when p≡ 3mod 4.
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BibTeX
@misc{miao2026szemeréditrotter,
title = {{The Szemerédi-Trotter Estimate in Finite Field with its Applications}},
author = {Changxing Miao and Rui Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35190},
eprint = {2609.35190},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35190},
}
RIS
TY - UNPB
TI - The Szemerédi-Trotter Estimate in Finite Field with its Applications
AU - Changxing Miao
AU - Rui Xie
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.35190
UR - https://arxiv.org/abs/2609.35190
AB - We prove a sharp Szemerédi-Trotter estimate ℐ(A,ℒ)≲ |A||ℒ|/p+|A|^2/3|ℒ|^2/3+|A|+|ℒ| for prime finite field 𝔽=𝔽_p by a new polynomial decomposition theorem. As applications, we first prove the sharp Furstenberg set estimate in 𝔽^2. Secondly, we improve sum-product estimate max{|A+A|,|A· A|}≳min{(p|A|)^1/2,|A|^5/4}, A⊂𝔽. Finally, we improve the Fourier restriction estimate R^*(2→α) holds for α>10/3 in 𝔽^3 when p≡ 3mod 4.
ER -