On the upper regularity dimensions of measures
Abstract
We study the \emph{upper regularity dimension} which describes the extremal local scaling behaviour of a measure and effectively quantifies the notion of \emph{doubling}. We conduct a thorough study of the upper regularity dimension, including its relationship with other concepts such as the Assouad dimension, the upper local dimension, the -spectrum and weak tangent measures. We also compute the upper regularity dimension explicitly in a number of important contexts including self-similar measures, self-affine measures, and measures on sequences.
Cite
@article{arxiv.1706.09340,
title = {On the upper regularity dimensions of measures},
author = {Jonathan M. Fraser and Douglas C. Howroyd},
journal= {arXiv preprint arXiv:1706.09340},
year = {2021}
}
Comments
22 pages, 4 figures. We corrected an error in Theorem 2.2 and provided a new example concerning weak tangent measures. Minor corrections were added in section 3.6.1, not changing any results. To appear in Indiana Univ. Math. J