On a radial projection conjecture and pinned directions in finite spaces

Abstract

We give upper bounds on the number of exceptional radial projections of arbitrary subsets of vector spaces over finite fields. Our bounds do not depend on the dimension of the ambient space. Let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over $\mathbb{F}_q$, let $k \in \{1,2,\ldots,d-1\}$, and let $E \subseteq \mathbb{F}_q^d$ be an arbitrary set of points. We prove two results. First, if $q^{k-1} < |E| \leq 100^{-1}q^{k}$, then the number of points $y$ such that the projection of $E$ from $y$ contains fewer than $50^{-1}|E|$ points is bounded above by $40q^k$. This establishes a conjecture of Lund, Pham, and Thu. Second, if $30q^{k} \leq |E| \leq q^{k+1}$, then the number of points $y$ such that the projection of $E$ from $y$ contains fewer than $M \leq 4^{-1}q^k$ points is bounded above by $300q^kM|E|^{-1}$. We also have an application to a pinned directions problem. Specifically, if $E\subset \mathbb{F}_q^d$ with $|E| > 30q^k$, then there is a point $y \in E$ such that the set of lines incident to $y$ and at least one other point of $E$ determines $q^k/4$ distinct slopes.