Existence of doubling measures via generalised nested cubes

TL;DR

This paper constructs generalized dyadic cubes in doubling metric spaces using ultrametric structures, providing a simple proof for the existence of doubling measures and demonstrating measures with full measure on low packing dimension sets.

Contribution

It introduces a new method to construct dyadic cubes in doubling spaces and proves the existence of doubling measures with specific dimensional properties.

Findings

Existence of doubling measures in complete doubling metric spaces.

Construction of doubling measures with full measure on sets of arbitrarily small packing dimension.

Simplified proof technique for the existence of doubling measures.

Abstract

Working on doubling metric spaces, we construct generalised dyadic cubes adapting ultrametric structure. If the space is complete, then the existence of such cubes and the mass distribution principle lead into a simple proof for the existence of doubling measures. As an application, we show that for each $\epsilon>0$ there is a doubling measure having full measure on a set of packing dimension at most $\epsilon$.