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arXiv · 2504.08127

Quasisymmetric rectifiability of uniformly disconnected sets

Jacob Honeycutt·Vyron Vellis
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Abstract

We prove that uniformly disconnected subsets of metric measure spaces with controlled geometry (complete, Ahlfors regular, supporting a Poincare inequality, and a mild topological condition) are contained in a quasisymmetric arc. This generalizes a result of MacManus in 1999 from Euclidean spaces to abstract metric setting. Along the way, we prove a geometric strengthening of the classical Denjoy-Riesz theorem in metric measure spaces. Finally, we prove that the complement of a uniformly disconnected set in such a metric space is uniform, quantitatively.

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Jacob Honeycutt, Vyron Vellis. 2025-04-10. Quasisymmetric rectifiability of uniformly disconnected sets. https://arxiv.org/abs/2504.08127

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