Dimension Estimates on Circular $(s,t)$-Furstenberg Sets

TL;DR

This paper establishes new lower bounds on the Hausdorff dimension of circular $(s,t)$-Furstenberg sets in the plane, extending prior results on circular Kakeya sets by Wolff.

Contribution

It provides the first comprehensive dimension estimates for circular $(s,t)$-Furstenberg sets, generalizing earlier work on circular Kakeya sets.

Findings

Hausdorff dimension at least max{t/3 + s, (2t+1)s - t}

extends Wolff's results on circular Kakeya sets

generalizes to all 0<s,t≤1

Abstract

In this paper, we show that circular $(s,t)$-Furstenberg sets in $\mathbb R^2$ have Hausdorff dimension at least $$\max\{\frac{t}3+s,(2t+1)s-t\} \text{ for all $0<s,t\le 1$}.$$ This result extends the previous dimension estimates on circular Kakeya sets by Wolff.