Harmonic Analysis

Distance sets corresponding to non-Euclidean norms

speaker: Izabela Laba (University of British Columbia)

abstract: Let X be the 2-dimensional plane equipped with a non-Euclidean norm in which the unit ball is a convex set K, and let S be a well-distributed subset of X. We address the question of how small the distance set of S in X can be, depending on properties of K. In particular, we prove that there is a well-distributed S whose distance set has bounded density if and only if K is a polygon with finitely many sides, all of which have algebraic slopes in some coordinate system. We also consider the "continuous" version of the problem, i.e. given a planar set E of positive Hausdorff dimension s, how does the dimension of its distance set in X depend on s and on the properties of K?

Results presented in this talk were obtained jointly with Alex Iosevich and with Sergei Konyagin.

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