On Falconer's distance set problem in the plane

Abstract

If $E\subset {\mathbb{R}}^2$ is a compact set of Hausdorff dimension greater than $5/4$, we prove that there is a point $x\in E$ so that the set of distances $\{|x-y|{\}}_{y\in E}$ has positive Lebesgue measure.

arXiv comment39 pages, 1 figure

References in corpus (7)

• Some connections between Falconer's distance set conjecture, and sets of Furstenburg type
• On the dimension and smoothness of radial projections
• On the distance sets of AD-regular sets
• Weighted restriction estimates and application to Falconer distance set problem
• New bounds on the dimensions of planar distance sets
• Pinned distance problem, slicing measures and local smoothing estimates
• Pointwise convergence of Schrödinger solutions and multilinear refined Strichartz estimates

Cited by in corpus (3)

• Bisector energy and pinned distances in positive characteristic
• An improved dimensional threshold for the angle problem
• Distribution of determinant of sum of matrices