On variants of the Furstenberg set problem

TL;DR

This paper investigates the packing and box dimensions of sets in Euclidean space that intersect a family of lines with specified dimensions, establishing sharp bounds and solving variants of the Furstenberg set problem.

Contribution

It provides sharp bounds for the packing and box dimensions of Furstenberg-type sets in Euclidean spaces, extending and solving variants of the classical problem.

Findings

Packing dimension of such sets is at least max{s, t/2}.

Sharp bounds are established for the box dimensions, at max{s, t+1-d}.

The results solve a variant of the Furstenberg set problem for packing dimension in 2D.

Abstract

Given an integer $d \geq 2$, $s \in (0,1]$, and $t \in [0,2(d-1)]$, suppose a set $X$ in $\mathbb{R}^d$ has the following property: there is a collection of lines of packing dimension $t$ such that every line from the collection intersects $X$ in a set of packing dimension at least $s$. We show that such sets must have packing dimension at least $\max\{s,t/2\}$ and that this bound is sharp. In particular, the special case $d=2$ solves a variant of the Furstenberg set problem for packing dimension. We also solve the upper and lower box dimension variants of the problem. In both of these cases the sharp threshold is $\max\{s,t+1-d\}$.