Boundedness of the density normalised Jones' square function does not imply $1$-rectifiability

TL;DR

This paper demonstrates that boundedness of the density-normalized Jones' square function does not imply 1-rectifiability by constructing a measure with bounded Jones' function but zero lower density, showing the converse does not hold.

Contribution

It provides a counterexample to the previously assumed implication between Jones' square function boundedness and 1-rectifiability.

Findings

Constructed a measure with bounded Jones' function but zero lower density almost everywhere.

Showed that bounded Jones' function does not imply 1-rectifiability.

Established that the converse of a known implication is false.

Abstract

Recently, M. Badger and R. Schul proved that for a $1$-rectifiable Radon measure $\mu$, the density weighted Jones' square function $$ J_{1}(x) = \mathop{\sum_{Q \in \mathcal{D}}}_{\ell(Q) \leq 1} \beta_{2,\mu}^{2}(3Q)\frac{\ell(Q)}{\mu(Q)} 1_{Q}(x) $$ is finite for $\mu$-a.e. $x$. Answering a question of Badger-Schul, we show that the converse is not true. Given $\epsilon > 0$, we construct a Radon probability measure on $[0,1]^{2} \subset \mathbb{R}^{2}$ with the properties that $J_{1}(x) \leq \epsilon$ for all $x \in \operatorname{spt} \mu$, but nevertheless the $1$-dimensional lower density of $\mu$ vanishes almost everywhere. In particular, $\mu$ is purely $1$-unrectifiable.