Sets which are not tube null and intersection properties of random measures

Abstract

We show that in ${\mathbb{R}}^d$ there are purely unrectifiable sets of Hausdorff (and even box counting) dimension $d-1$ which are not tube null, settling a question of Carbery, Soria and Vargas, and improving a number of results by the same authors and by Carbery. Our method extends also to "convex tube null sets", establishing a contrast with a theorem of Alberti, Csörnyei and Preiss on Lipschitz-null sets. The sets we construct are random, and the proofs depend on intersection properties of certain random fractal measures with curves.

24 pages. Referees comments incorporated. JLMS to appear

References in corpus (1)

• The dimension of projections of fractal percolations

Cited by in corpus (2)

• Distribution of random Cantor sets on Tubes
• On the tube-occupancy of sets in ${\mathbb{R}}^d$