Restricted projections in positive characteristic via Fourier extension and restriction estimates

Abstract

Let $d\ge 3$ 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 $X_{\pi }$ be the set of lines through the origin intersecting the slice $\pi \cap S^{d-1}$, where $\pi =\{x_d=\lambda \}$ and $S^{d-1}=\{x:\Vert x\Vert =1\}$. For $E\subset {\mathbb{F}}_q^d$ and $N\ge 1$, 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 $|T_1|$ and $|T_2|$, with separate analyses for the cases $\lambda \in \{0,\pm 1\}$. The bounds exhibit arithmetic-geometric dichotomies absent in the full Grassmannian: the quadratic character of ${\lambda }^2-1$ 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 $\mathbf{y}\in \pi \cap S^{d-1}$ such that the pinned dot-product sets $\{\mathbf{y}\cdot \mathbf{x}:\mathbf{x}\in E\}$ have cardinality $\Omega (q)$. 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).

V2: typos corrected