On a radial projection conjecture and pinned directions in finite spaces

TL;DR

The paper studies radial projections in finite field vector spaces and bounds the number of exceptional projections for arbitrary finite-field point sets, achieving dimension-free estimates. The authors introduce a dimension-reduction technique via a random (d−k−1)-dimensional subspace that approximately preserves the sizes of the original set and the exceptional set, enabling reduction to a lower-dimensional setting where existing bounds apply. They prove two main results: (i) for modestly sized E, the number of exceptional projections is at most a constant times q^k, and (ii) for larger E, one gains a bound depending on M and |E|; they also deduce a pinned-directions consequence, guaranteeing many slopes through some point of E when |E| is large. The approach connects radial projections to pinned-direction problems and complements finite-field distance-set results, with ties to continuous analogs in the literature.

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.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• proof
• Lemma 3.1
• proof
• Lemma 3.2: LPT
• Theorem 3.3: LPT
• proof : Proof of \ref{['th:fullDimLargeESC']}
• Theorem 3.4: LPT