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- W4386501877 abstract "We prove that for all integers $2leq mleq d-1$, there exists doubling measures on $mathbb{R}^d$ with full support that are $m$-rectifiable and purely $(m-1)$-unrectifiable in the sense of Federer (i.e. without assuming $mullmathcal{H}^m$). The corresponding result for 1-rectifiable measures is originally due to Garnett, Killip, and Schul (2010). Our construction of higher-dimensional Lipschitz images is informed by a simple observation about square packing in the plane: $N$ axis-parallel squares of side length $s$ pack inside of a square of side length $lceil N^{1/2}rceil s$. The approach is robust and when combined with standard metric geometry techniques allows for constructions in complete Ahlfors regular metric spaces. One consequence of the main theorem is that for each $min{2,3,4}$ and $s<m$, there exist doubling measures $mu$ on the Heisenberg group $mathbb{H}^1$ and Lipschitz maps $f:Esubsetmathbb{R}^mrightarrowmathbb{H}^1$ such that $mullmathcal{H}^{s-epsilon}$ for all $epsilon>0$, $f(E)$ has Hausdorff dimension $s$, and $mu(f(E))>0$. This is striking, because $mathcal{H}^m(f(E))=0$ for every Lipschitz map $f:Esubsetmathbb{R}^mrightarrowmathbb{H}^1$ by a theorem of Ambrosio and Kirchheim (2000). Another application of the square packing construction is that every compact metric space $mathbb{X}$ of Assouad dimension strictly less than $m$ is a Lipschitz image of a compact set $Esubset[0,1]^m$. Of independent interest, we record the existence of doubling measures on complete Ahlfors regular metric spaces with prescribed lower and upper Hausdorff and packing dimensions." @default.
- W4386501877 created "2023-09-08" @default.
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- W4386501877 date "2023-09-03" @default.
- W4386501877 modified "2023-10-16" @default.
- W4386501877 title "Square packings and rectifiable doubling measures" @default.
- W4386501877 doi "https://doi.org/10.48550/arxiv.2309.01283" @default.
- W4386501877 hasPublicationYear "2023" @default.
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