Furstenberg set problem and exceptional set estimate in prime fields: dimension two implies higher dimensions

TL;DR

This paper explores exceptional set estimates and Furstenberg-type problems in prime fields, demonstrating that results in two dimensions extend to higher dimensions, refining projection theorems in finite field contexts.

Contribution

It establishes that two-dimensional results in prime fields imply analogous higher-dimensional results, connecting Furstenberg set problems with projection theorems.

Findings

Two-dimensional exceptional set estimates imply higher-dimensional results.

Refinement of Marstrand's projection theorem in prime fields.

Connection between Furstenberg problems and dimension theory in finite fields.

Abstract

In this paper, we study in prime fields the exceptional set estimates, which can be viewed as a refinement of Marstrand's orthogonal projection theorem. Additionally, we address a Furstenberg-type problem, which is closely related. It is shown that the two-dimensional result implies all higher-dimensional results in the prime-field setting.