PreprintCC BY 4.0arXiv
A combinatorial approach to point-flat incidences over finite fields
Abstract
We establish new bounds for incidences between a point set P and a family L of n-flats in PG(n+d,q). For fixed dimensions, our bound on the incidence discrepancy has an explicit piecewise-linear exponent in logq|L|, improving the classical estimate of Haemers and the Kong-Tamo bound in specified ranges of the number of flats. Matching constructions establish sharpness up to constant factors in several parameter ranges. The proof is combinatorial and avoids spectral and Fourier analytic methods. As applications, we obtain improved estimates for rich flats and exceptional orthogonal projections, together with stronger lower bounds for Furstenberg sets in certain ranges where the fraction of prescribed directions is small.
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BibTeX
@misc{lund2026combinatorial,
title = {{A combinatorial approach to point-flat incidences over finite fields}},
author = {Ben Lund and Tao Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33365},
eprint = {2609.33365},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33365},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A combinatorial approach to point-flat incidences over finite fields
AU - Ben Lund
AU - Tao Zhang
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33365
UR - https://arxiv.org/abs/2609.33365
AB - We establish new bounds for incidences between a point set P and a family L of n-flats in PG(n+d,q). For fixed dimensions, our bound on the incidence discrepancy has an explicit piecewise-linear exponent in log_q|L|, improving the classical estimate of Haemers and the Kong-Tamo bound in specified ranges of the number of flats. Matching constructions establish sharpness up to constant factors in several parameter ranges. The proof is combinatorial and avoids spectral and Fourier analytic methods. As applications, we obtain improved estimates for rich flats and exceptional orthogonal projections, together with stronger lower bounds for Furstenberg sets in certain ranges where the fraction of prescribed directions is small.
ER -