arXiv subjects

Douglas C. Howroyd

Publications and source records attributed to Douglas C. Howroyd.

5 recordsLinked to original sources

arXiv2021

On the Hausdorff dimension of microsets

We investigate how the Hausdorff dimensions of microsets are related to the dimensions of the original set. It is known that the maximal dimension of a microset is the Assouad dimension of the set. We prove that the lower dimension can analogously be obtained as the minimal dimension of a microset. In particular, the maximum and minimum exist. We also show that for an arbitrary $\mathcal{F}_σ$ set $Δ\subseteq [0,d]$ containing its infimum and supremum there is a compact set in $[0,1]^d$ for which the set of Hausdorff dimensions attained by its microsets is exactly equal to the set $Δ$. Our work is motivated by the general programme of determining what geometric information about a set can be determined at the level of tangents.

math.MG↗

arXiv2019

Regularity dimensions: quantifying doubling and uniform perfectness

We study the upper and lower regularity dimensions in relation to the notions of doubling and uniformly perfect. These two regularity properties are closely related which is quantified thanks to the regularity dimensions. The regularity dimensions of pushforward measures onto graphs of Brownian motion are calculated, similarly for pushforwards with respect to quasisymmetric homeomorphisms. We finish by introducing an application to Diophantine approximation in the setting of Kleinian groups.

math.MG↗

arXiv2019

On the upper regularity dimensions of measures

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 $L^q$-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.

math.MG↗

arXiv2018

Dimension growth for iterated sumsets

We study dimensions of sumsets and iterated sumsets and provide natural conditions which guarantee that a set $F \subseteq \mathbb{R}$ satisfies $\overline{\dim}_\text{B} F+F > \overline{\dim}_\text{B} F$ or even $\dim_\text{H} n F \to 1$. Our results apply to, for example, all uniformly perfect sets, which include Ahlfors-David regular sets. Our proofs rely on Hochman's inverse theorem for entropy and the Assouad and lower dimensions play a critical role. We give several applications of our results including an Erdős-Volkmann type theorem for semigroups and new lower bounds for the box dimensions of distance sets for sets with small dimension.

math.MG↗

arXiv2016

Assouad type dimensions for self-affine sponges

We study the Assouad and lower dimensions of self-affine sponges; the higher dimensional analogue of the planar self-affine carpets of Bedford and McMullen. Our techniques involve the weak tangents of Mackay and Tyson as well as regularity properties of doubling measures in the context studied by Bylund and Gudayol.

math.MG↗