Furstenberg sets estimate in the plane

Abstract: A $(s,t)$-Furstenberg set is a set $E$ in the plane with the following property: there exists a $t$-dim family of lines such that each line intersects $E$ in a $\geq s$--dimensional set. An unpublished conjecture of Furstenberg states that any $(s,1)$-Furstenberg set has dimension at least $(3s+1)/2$. The Furstenberg set problem can be viewed as a natural generalization of Davies's result that a Kakeya set in the plane (a set that contains a line segment in any direction) has dimension 2.

We will survey a sequence of results by Orponen, Shmerkin, and a joint work with Ren that lead to the solution of the Furstenberg set conjecture in the plane: any $(s,t)$-Furstenberg set has Hausdorff dimension at least $\min \{s+t, (3s+t)/2, s+1\}$.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdifferential geometryfunctional analysisgeometric topologyspectral theory

Audience: researchers in the topic

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Geometric and functional inequalities and applications

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