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 $f \colon \mathbb{R} \to \mathbb{R}$ 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 $\mathbb{R}$.