On the tube-occupancy of sets in ${\mathbb{R}}^d$

Abstract

Call a pair $(s,t)\in [0,d]\times [0,d]$ admissible, if there exists a compact set $K\subset {\mathbb{R}}^d$ and a constant $C>0$ such that $0<{\mathcal{H}}^s(K)<\infty$, and $${\mathcal{H}}_s(K\cap T)\le Cw(T{)}^t$$ for all tubes $T\subset {\mathbb{R}}^d$ of width $w(T)$. The purpose of this paper is to show that all pairs $(s,s)$ with $s<1$ are admissible. Combined with previous results, this settles a question of A. Carbery.

15 pages. v2: referee comments incorporated, to appear in IMRN

References in corpus (1)

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

Cited by in corpus (2)

• Distribution of random Cantor sets on Tubes
• Covering the Sierpiński carpet with tubes