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

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

2012-04-26

arxiv.org/abs/1204.5883v1

Mathematics

Classical Analysis and ODEs

Scientific paper

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, Cs\"{o}rnyei 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.

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