On the Hausdorff dimension of circular Furstenberg sets

Abstract

For $0\le s\le 1$ and $0\le t\le 3$, 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 ${\dim }_{\mathrm{H}}\mathcal{S}\ge t$ such that $${\dim }_{\mathrm{H}}(F\cap S)\ge s,S\in \mathcal{S}.$$ We prove that if $0\le t\le s\le 1$, then every circular $(s,t)$-Furstenberg set $F\subset {\mathbb{R}}^2$ has Hausdorff dimension ${\dim }_{\mathrm{H}}F\ge s+t$. The case $s=1$ follows from earlier work of Wolff on circular Kakeya sets.

v2: 83 pages, 5 figures. This version is published by Discrete Analysis