On the Hausdorff dimension of circular Furstenberg sets

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) 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.