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

Abstract

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

4 pages, not intended for publication