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

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.

arXiv comment16 pages, the upper bound part rewritten completely and the title changed slightly

References in corpus (1)

• On the size of Kakeya sets in finite fields

Cited by in corpus (1)

• Furstenberg sets and Furstenberg schemes over finite fields