Analysis Seminar

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Furstenberg sets estimate in the plane

Speaker: Hong Wang, New York University

Location: Warren Weaver Hall 1302

Date: Thursday, September 12, 2024, 11 a.m.

Synopsis:

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 An unpublished conjecture of Furstenberg states that any (s,1)-Furstenberg set has dimension at least \((3s+1)/2\).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 \(\min \{s+t, (3s+t)/2, s+1\}\).