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arXiv · 1705.04961

Every locally finite Borel measure on $\mathbb{R}$ has conformal dimension zero

Tuomas Orponen
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Abstract

A result of P. Tukia from 1989 says that Lebesgue measure on $\mathbb{R}$ has conformal dimension zero: for every $ε> 0$, there is a Borel set $G \subset \mathbb{R}$ of full Lebesgue measure, and a quasisymmetric homeomorphism $f \colon \mathbb{R} \to \mathbb{R}$ such that $\dim_{\mathrm{H}} f(G) < ε$. In this short note, I show that the same is true for every locally finite Borel measure on $\mathbb{R}$.

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Tuomas Orponen. 2017-05-14. Every locally finite Borel measure on $\mathbb{R}$ has conformal dimension zero. https://arxiv.org/abs/1705.04961

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