JournalsrmiVol. 37, No. 5pp. 1953–1968

Falconer’s (K,d)(K, d) distance set conjecture can fail for strictly convex sets KK in Rd\mathbb R^d

  • Christopher J. Bishop

    Stony Brook University, USA
  • Hindy Drillick

    Columbia University, New York, USA
  • Dimitrios Ntalampekos

    Stony Brook University, USA
Falconer’s $(K, d)$ distance set conjecture can fail for strictly convex sets $K$ in $\mathbb R^d$ cover
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Abstract

For any norm on Rd\mathbb{R}^d with countably many extreme points, we prove that there is a set ERdE \subset \mathbb{R}^d of Hausdorff dimension dd whose distance set with respect to this norm has zero linear measure. This was previously known only for norms associated to certain finite polygons in R2\mathbb{R}^2. Similar examples exist for norms that are very well approximated by polyhedral norms, including some examples where the unit ball is strictly convex and has C1C^1 boundary.

Cite this article

Christopher J. Bishop, Hindy Drillick, Dimitrios Ntalampekos, Falconer’s (K,d)(K, d) distance set conjecture can fail for strictly convex sets KK in Rd\mathbb R^d. Rev. Mat. Iberoam. 37 (2021), no. 5, pp. 1953–1968

DOI 10.4171/RMI/1254