On configurations where the Loomis-Whitney inequality is nearly sharp and applications to the Furstenberg set problem

TL;DR

This paper investigates the Furstenberg set problem in high dimensions, providing new bounds and insights by analyzing near-sharp configurations of the Loomis-Whitney inequality, with applications to finite fields and Euclidean spaces.

Contribution

It introduces a theorem on near-sharp Loomis-Whitney configurations that reduces high-dimensional problems to two dimensions, improving bounds for the Furstenberg set problem.

Findings

Established upper bounds for Furstenberg sets in real and finite fields.

Improved the critical exponent by a factor of /n^2 in finite fields.

Linked near-sharp Loomis-Whitney configurations to dimension reduction.

Abstract

In this paper, we consider the so-called "Furstenberg set problem" in high dimensions. First, following Wolff's work on the two dimensional real case, we provide "reasonable" upper bounds for the problem for $\mathbb{R}$ or $\mathbb{F}_p$. Next we study the "critical" case and improve the "trivial" exponent by $\Omega (\frac{1}{n^2})$ for $\mathbb{F}_p^n$. Our key tool to obtain this lower bound is a theorem about how things behave when the Loomis-Whitney inequality is nearly sharp, as it helps us to reduce the problem down to dimension two.