On the Hausdorff dimension of circular Furstenberg sets

TL;DR

This paper establishes a lower bound on the Hausdorff dimension of circular Furstenberg sets, showing that such sets have dimension at least the sum of parameters s and t, extending previous results on circular Kakeya sets.

Contribution

It proves a new lower bound for the Hausdorff dimension of circular Furstenberg sets in the case where 0 ≤ t ≤ s ≤ 1, generalizing earlier work on circular Kakeya sets.

Findings

Hausdorff dimension of circular Furstenberg sets is at least s + t for 0 ≤ t ≤ s ≤ 1

Extends previous results on circular Kakeya sets to a broader class of Furstenberg sets

Provides a dimension estimate linking the size of the family of circles and the intersection dimension

Abstract

For $0 \leq s \leq 1$ and $0 \leq t \leq 3$, a set $F \subset \mathbb{R}^{2}$ is called a circular $(s,t)$-Furstenberg set if there exists a family of circles $\mathcal{S}$ of Hausdorff dimension $\dim_{\mathrm{H}} \mathcal{S} \geq t$ such that $$\dim_{\mathrm{H}} (F \cap S) \geq s, \qquad S \in \mathcal{S}.$$ We prove that if $0 \leq t \leq s \leq 1$, then every circular $(s,t)$-Furstenberg set $F \subset \mathbb{R}^{2}$ has Hausdorff dimension $\dim_{\mathrm{H}} F \geq s + t$. The case $s = 1$ follows from earlier work of Wolff on circular Kakeya sets.