Square packings and rectifiable doubling measures
Matthew Badger, Raanan Schul

TL;DR
This paper constructs doubling measures on Euclidean and metric spaces that are rectifiable in certain dimensions but purely unrectifiable in others, using a novel square packing approach to analyze Lipschitz images.
Contribution
It introduces a new method based on square packing for constructing higher-dimensional Lipschitz images with specific measure and rectifiability properties, extending to general metric spaces.
Findings
Existence of doubling measures that are m-rectifiable and purely (m-1)-unrectifiable.
Construction of Lipschitz images with prescribed Hausdorff dimension and measure properties in Heisenberg group.
Every compact metric space with Assouad dimension less than m is a Lipschitz image of a subset of [0,1]^m.
Abstract
We prove that for all integers , there exists doubling measures on with full support that are -rectifiable and purely -unrectifiable in the sense of Federer (i.e. without assuming ). 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: axis-parallel squares of side length pack inside of a square of side length . 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 and , there exist doubling measures on the Heisenberg group and…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Mathematical Dynamics and Fractals · Limits and Structures in Graph Theory
