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Separated sets and the Falconer conjecture for polygonal norms

Sergei Konyagin·Izabella Laba
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Abstract

The Falconer conjecture asserts that if E is a planar set with Hausdorff dimension strictly greater than 1, then its Euclidean distance set has positive one-dimensional Lebesgue measure. We discuss the analogous question with the Euclidean distance replaced by non-Euclidean norms in which the unit ball is a polygon with 2K sides. We prove that for any such norm there is a set of Hausdorff dimension K/(K-1) whose distance set has Lebesgue measure 0.

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Sergei Konyagin, Izabella Laba. 2004-07-28. Separated sets and the Falconer conjecture for polygonal norms. https://arxiv.org/abs/math/0407503

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