On some variants of the Kakeya problem

Lawrence Kolasa and Thomas Wolff

Vol. 190 (1999), No. 1, 111–154

DOI: 10.2140/pjm.1999.190.111

Abstract

We study the question of lower bounds for the Hausdorff dimension of a set in Rⁿ containing spheres of every radius. If n ≥ 3 then such a set must have dimension n. If n = 2 then it must have dimension at least 11/6. We also study the analogous maximal function problem and related problem of Besicovitch sets with an axis of symmetry.

Milestones

Received: 22 January 1996

Revised: 20 May 1997

Published: 1 September 1999