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arXiv · 1204.5883

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

Pablo Shmerkin·Ville Suomala
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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.

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Pablo Shmerkin, Ville Suomala. 2014-11-26. Sets which are not tube null and intersection properties of random measures. https://doi.org/10.1112/jlms%2Fjdu083

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