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

Christopher J. Bishop; Hindy Drillick; Dimitrios Ntalampekos

doi:10.4171/rmi/1254

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

Falconer’s $(K, d)$ distance set conjecture can fail for strictly convex sets $K$ in $R^{d}$

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 $R^{d}$ with countably many extreme points, we prove that there is a set $E \subset R^{d}$ of Hausdorff dimension $d$ 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 $R^{2}$ . 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 $C^{1}$ boundary.

Cite this article

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

DOI 10.4171/RMI/1254