Restricted projections in positive characteristic via Fourier extension and restriction estimates
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915669025488896 |
|---|---|
| author | Ham, Le Quang Hoang, Do Trong Hung, Le Quang Koh, Doowon Pham, Thang |
| author_facet | Ham, Le Quang Hoang, Do Trong Hung, Le Quang Koh, Doowon Pham, Thang |
| contents | Let $d\ge3$ and $\mathbb{F}_q^{\,d}$ be the $d$-dimensional vector space over a finite field of order $q$, where $q$ is an odd prime power. Let $X_π$ be the set of lines through the origin intersecting the slice $π\cap S^{d-1}$, where $π=\{x_d=λ\}$ and $S^{d-1}=\{x:\|x\|=1\}$. For $E\subset\mathbb{F}_q^{\,d}$ and $N\ge1$, we study the exceptional sets \[ T_1(X_π,E,N)=\bigl\{V\in X_π:\ |π_V(E)|\le N\bigr\},\qquad T_2(X_π,E,N)=\bigl\{V\in X_π:\ |π_{V^\perp}(E)|\le N\bigr\}, \] with their respective natural ranges of $N$. Using discrete Fourier analysis together with restriction/extension estimates for cone and sphere-type quadrics over finite fields, we obtain sharp upper bounds (up to constant factors) for $\lvert T_1\rvert$ and $\lvert T_2\rvert$, with separate analyses for the cases $λ\in \{0, \pm 1\}$. The bounds exhibit arithmetic-geometric dichotomies absent in the full Grassmannian: the quadratic character of $λ^{2}-1$ and the parity of $d$ determine the size of the exceptional sets. As an application, when $|E|\ge q$, there exists a positive proportion of elements $\mathbf{y}\in π\cap S^{d-1}$ such that the pinned dot-product sets $\{\mathbf{y}\cdot \mathbf{x}\colon \mathbf{x}\in E\}$ have cardinality $Ω(q)$. We further study analogous families arising from the spheres of radii $0$ and $-1$, and, by combining the results, recover the known estimates for projections over the full Grassmannian, complementing a result of Chen (2018). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_05522 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Restricted projections in positive characteristic via Fourier extension and restriction estimates Ham, Le Quang Hoang, Do Trong Hung, Le Quang Koh, Doowon Pham, Thang Combinatorics Classical Analysis and ODEs Number Theory Let $d\ge3$ and $\mathbb{F}_q^{\,d}$ be the $d$-dimensional vector space over a finite field of order $q$, where $q$ is an odd prime power. Let $X_π$ be the set of lines through the origin intersecting the slice $π\cap S^{d-1}$, where $π=\{x_d=λ\}$ and $S^{d-1}=\{x:\|x\|=1\}$. For $E\subset\mathbb{F}_q^{\,d}$ and $N\ge1$, we study the exceptional sets \[ T_1(X_π,E,N)=\bigl\{V\in X_π:\ |π_V(E)|\le N\bigr\},\qquad T_2(X_π,E,N)=\bigl\{V\in X_π:\ |π_{V^\perp}(E)|\le N\bigr\}, \] with their respective natural ranges of $N$. Using discrete Fourier analysis together with restriction/extension estimates for cone and sphere-type quadrics over finite fields, we obtain sharp upper bounds (up to constant factors) for $\lvert T_1\rvert$ and $\lvert T_2\rvert$, with separate analyses for the cases $λ\in \{0, \pm 1\}$. The bounds exhibit arithmetic-geometric dichotomies absent in the full Grassmannian: the quadratic character of $λ^{2}-1$ and the parity of $d$ determine the size of the exceptional sets. As an application, when $|E|\ge q$, there exists a positive proportion of elements $\mathbf{y}\in π\cap S^{d-1}$ such that the pinned dot-product sets $\{\mathbf{y}\cdot \mathbf{x}\colon \mathbf{x}\in E\}$ have cardinality $Ω(q)$. We further study analogous families arising from the spheres of radii $0$ and $-1$, and, by combining the results, recover the known estimates for projections over the full Grassmannian, complementing a result of Chen (2018). |
| title | Restricted projections in positive characteristic via Fourier extension and restriction estimates |
| topic | Combinatorics Classical Analysis and ODEs Number Theory |
| url | https://arxiv.org/abs/2510.05522 |