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

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.