On the Hausdorff dimension of circular Furstenberg sets

Abstract

On the Hausdorff dimension of circular Furstenberg sets, Discrete Analysis 2024:18, 83 pp. In the late 1990s, Wolff proved the following result. Suppose that the set $E\subset \mathbb R^2$ contains circles centered at all points of a Borel set with Hausdorff dimension at least $t$. Then the Hausdorff dimension of $E$ is at least $1+t$. This may be viewed as a "Kakeya-type" result for circles, showing that one cannot pack too many circles in a small set. Now consider the following more general problem. We say that a family of circles $\mathcal{S}$ in $\mathbb R^2$ is $t$-dimensional if $t$ is the Hausdorff dimension of the set of pairs $(x,r)\in\mathbb R^2\times[0,\infty)$ such that $x$ and $r$ are the center and radius, respectively, of some circle in $\mathcal{S}$. A set $F\subset\mathbb R^2$ is called a _circular $(s,t)$-Furstenberg set_ if there is a $t$-dimensional family of circles $\mathcal{S}$ such that $F$ intersects every circle $S\in\mathcal{S}$ in a set of Hausdorff dimension at least $s$. What can we say about lower bounds on the dimension of circular $(s,t)$-Furstenberg sets? This also generalizes another well-known question in analysis: the _linear_ $(s,t)$-Furstenberg set problem, which has a similar statement, but with circles replaced by lines. The linear question was posed by Wolff in connection with his work on the Kakeya problem, and resolved recently by Orponen-Shmerkin and Ren-Wang (the two papers considered different cases of the problem). Using a projective transformation, one can map the linear problem to a special case of the circular problem, hence the latter is more general. In the present paper, the authors prove that a circular $(s,t)$-Furstenberg set with $0\leq t\leq s\leq 1$ has Hausdorff dimension at least $s+t$. The case $t=1$ follows from the aforementioned work of Wolff; while Wolff assumed that the set $E$ contains entire circles rather than their 1-dimensional subsets, his proof is robust enough to cover this case. The case $t<1$ is new and requires sophisticated arguments counting tangencies between circles. After this paper was first posted, Zahl ([arXiv:2307.05894](https://arxiv.org/abs/2307.05894)) proved a significant generalization of its main theorem. He gave the same lower bound $\dim_H(F)\geq s+t$, but for a broader class of $(s,t)$-Furstenberg sets that allows more general curves instead of circles. His proof is different and based on "broad-narrow" arguments closer to decoupling, rather than the bilinearization method developed in the article here.