Exceptional projections in finite fields: Fourier analytic bounds and incidence geometry
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915254598893568 |
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| author | Fraser, Jonathan M. Rakhmonov, Firdavs |
| author_facet | Fraser, Jonathan M. Rakhmonov, Firdavs |
| contents | We consider the problem of bounding the number of exceptional projections (projections which are smaller than typical) of a subset of a vector space over a finite field onto subspaces. We establish bounds that depend on $L^p$ estimates for the Fourier transform, improving various known bounds for sets with sufficiently good Fourier analytic properties. The special case $p=2$ recovers a recent result of Bright and Gan (following Chen), which established the finite field analogue of Peres--Schlag's bounds from the continuous setting. We prove several auxiliary results of independent interest, including a character sum identity for subspaces (solving a problem of Chen) and a full generalization of Plancherel's theorem for subspaces. These auxiliary results also have applications in affine incidence geometry, that is, the problem of estimating the number of incidences between a set of points and a set of affine $k$-planes. We present a novel and direct proof of a well-known result in this area that avoids the use of spectral graph theory, and we provide simple examples demonstrating that these estimates are sharp up to constants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_15072 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exceptional projections in finite fields: Fourier analytic bounds and incidence geometry Fraser, Jonathan M. Rakhmonov, Firdavs Combinatorics Classical Analysis and ODEs primary: 05B25, 42B10, 11B30, secondary: 51A05, 28A78 We consider the problem of bounding the number of exceptional projections (projections which are smaller than typical) of a subset of a vector space over a finite field onto subspaces. We establish bounds that depend on $L^p$ estimates for the Fourier transform, improving various known bounds for sets with sufficiently good Fourier analytic properties. The special case $p=2$ recovers a recent result of Bright and Gan (following Chen), which established the finite field analogue of Peres--Schlag's bounds from the continuous setting. We prove several auxiliary results of independent interest, including a character sum identity for subspaces (solving a problem of Chen) and a full generalization of Plancherel's theorem for subspaces. These auxiliary results also have applications in affine incidence geometry, that is, the problem of estimating the number of incidences between a set of points and a set of affine $k$-planes. We present a novel and direct proof of a well-known result in this area that avoids the use of spectral graph theory, and we provide simple examples demonstrating that these estimates are sharp up to constants. |
| title | Exceptional projections in finite fields: Fourier analytic bounds and incidence geometry |
| topic | Combinatorics Classical Analysis and ODEs primary: 05B25, 42B10, 11B30, secondary: 51A05, 28A78 |
| url | https://arxiv.org/abs/2503.15072 |