Radial projection theorems in finite spaces

TL;DR

This paper extends radial projection theorems to finite field spaces, providing new results that sometimes surpass continuous analogs and solving a conjecture about exceptional sets in finite fields.

Contribution

It generalizes and strengthens existing radial projection results in finite fields, including solving a conjecture on exceptional sets for sets of certain dimensions.

Findings

Finite field radial projection theorems established

Results sometimes stronger than continuous case

Solved a conjecture on exceptional sets in finite fields

Abstract

Motivated by recent results on radial projections and applications to the celebrated Falconer distance problem, we study radial projections in the setting of finite fields. More precisely, we extend results due to Mattila and Orponen (2016), Orponen (2018), and Liu (2020) to finite spaces. In some cases, our results are stronger than the corresponding results in the continuous setting. In particular, we solve the finite field analog of a conjecture due to Liu and Orponen on the exceptional set of radial projections of a set of dimension between $d-2$ and $d-1$.