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

## TL;DR

This paper extends Tukia's 1989 result by proving that all locally finite Borel measures on the real line can be transformed via quasisymmetric homeomorphisms to sets with arbitrarily small Hausdorff dimension, showing their conformal dimension is zero.

## Contribution

It generalizes Tukia's result from Lebesgue measure to all locally finite Borel measures on
, establishing that they all have conformal dimension zero.

## Key findings

- All locally finite Borel measures on
have conformal dimension zero.
- Existence of quasisymmetric homeomorphisms reducing measure supports to arbitrarily small Hausdorff dimension.
- Generalization of Tukia's 1989 result to broader class of measures.

## 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 $\mathbb{R}$.