Sharp Furstenberg sets estimate in the plane

Abstract: Fix two real numbers $s \in (0, 1]$, $t \in (0, 2]$. A set $E \subset \mathbb{R}^2$ is a $(s, t)$-Furstenberg set if there exists a set of lines $\mathcal{L}$ with Hausdorff dimension $t$ such that for each line $\ell \in \mathcal{L}$, we have $\dim_H (E \cap \ell) \ge s$. (For example, a Kakeya set in $\mathbb{R}^2$ is a special case of a $(1, 1)$-Furstenberg set.) The Furstenberg sets problem asks for the minimum possible Hausdorff dimension of a $(s, t)$-Furstenberg set for any given pair of $s, t$. In this talk, I will illustrate the rich theory linking this problem to the discretized sum-product problem, orthogonal projections, and the high-low method in Fourier analysis. Joint works with Yuqiu Fu and Hong Wang.

analysis of PDEsclassical analysis and ODEsfunctional analysis

Audience: researchers in the topic

OARS Online Analysis Research Seminar

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