Conformal dimension: Cantor sets and moduli

TL;DR

This paper investigates conditions under which spaces are minimal for conformal dimension, providing new examples of sets with zero length and conformal dimension 1, and establishing criteria involving measure moduli.

Contribution

It introduces new conditions for minimality in conformal dimension, including examples of zero-length sets with conformal dimension 1 and criteria based on measure moduli.

Findings

Existence of zero-length sets with conformal dimension 1

New sufficient conditions for minimality involving measure moduli

Many zero-length sets yield minimal product spaces for conformal dimension

Abstract

In this paper we give several conditions for a space to be minimal for conformal dimension. We show that there are sets of zero length and conformal dimension 1 thus answering a question of Bishop and Tyson. Another sufficient condition for minimality is given in terms of a modulus of a system of measures in the sense of Fuglede \cite{Fug}. It implies in particular that there are many sets $E\subset\mathbb{R}$ of zero length such that $X\times Y$ is minimal for conformal dimension for every compact $Y$.