Spread Furstenberg Sets

TL;DR

This paper establishes new lower bounds on the dimension of spread Furstenberg sets in high-dimensional Euclidean spaces, extending classical geometric measure theory results to a broader setting involving flats and their intersections.

Contribution

The paper introduces novel bounds for high-dimensional spread Furstenberg sets, generalizing previous results and employing methods inspired by finite field geometry.

Findings

Derived explicit lower bounds for the dimension of spread Furstenberg sets.

Extended classical Furstenberg set results to high-dimensional flats.

Utilized techniques inspired by finite field geometry and prior work of Dvir and Lund.

Abstract

We obtain new bounds for (a variant of) the Furstenberg set problem for high dimensional flats over $\mathbb{R}^n$. In particular, let $F\subset \mathbb{R}^n$, $1\leq k \leq n-1$, $s\in (0,k]$, and $t\in (0,k(n-k)]$. We say that $F$ is a $(s,t;k)$-spread Furstenberg set if there exists a $t$-dimensional set of subspaces $\mathcal P \subset \mathcal G(n,k)$ such that for all $P\in \mathcal P$, there exists a translation vector $a_P \in \mathbb{R}^n$ such that $\dim(F\cap (P + a_P)) \geq s$. We show that given $k \geq k_0 +1$ (where $k_0:= k_0(n)$ is sufficiently large) and $s>k_0$, every $(s,t;k)$-spread Furstenberg set $F$ in $\mathbb{R}^n$ satisfies \[ \dim F \geq n-k + s - \frac{k(n-k) - t}{\lceil s\rceil - k_0 +1 }. \] Our methodology is motivated by the work of the second author, Dvir, and Lund over finite fields.