arXiv subjects

Richard A. Howat

Publications and source records attributed to Richard A. Howat.

3 recordsLinked to original sources

arXiv2026

Cut and project schemes in the Poincar\'e disc: From cocompact Fuchsian groups to chaotic Delone sets

A question raised by Davies et al [Phys. Rev. Lett. 131, 2023] is: "Can developing new cut and project models, where the lattice is not square or the curve is non-linear, generate better performing graded metamaterials?" In this article, we study a natural construction of such a cut and project scheme, namely, cut and project schemes in relation to cocompact Fuchsian groups acting on the Poincar\'e disc model of hyperbolic space. We present a condition on the fundamental domain (a hyperbolic polygon) of the group so that the resulting cut and project set $S \subset \mathbb{R}$ is a chaotic Delone set. We also investigate the set of tile lengths of $S$, namely $\mathcal{L}_{S} = \{ z - y : z,y \in S, \, z > y \; \text{and} \; (y,z) \cap S = \emptyset \}$, and show that this set is countably infinite. Finally, we apply our results to cocompact Fuchsian triangle groups and show that the resulting cut and project sets are chaotic Delone, complementing and extending the work of L\'{o}pez et al. [Discrete Contin. Dyn. Syst. 41, 2021].

math.DS↗

arXiv2026

Affine thickness: Patterns and a Gap Lemma

A new notion of thickness for subsets of $B[0,1]\subset \mathbb{R}^n$ called affine thickness is defined; this notion of thickness is a generalisation of Falconer-Yavicoli thickness and is adapted to be used in the study of certain sets with affine cut outs. Thick sets are proven to be winning for the matrix potential game introduced in (arXiv:2508.11577) and as an application we can prove that for a thick set, there exists $M\in\mathbb{N}$ depending on the thickness of the set, such that the set contains a homothetic copy of every finite set with at most $M$ elements. Additionally, the author provides a counter-example to the gap lemma in $\mathbb{R}^n$ ($n\geq 2$) for Falconer-Yavicoli thickness, stated in (Math. Z., 2022) proving this result does not hold in the generality stated. We go on to provide a gap lemma for affine thickness in $\mathbb{R}^n$ (for $n\geq 2$) under additional conditions to the classical Newhouse gap lemma.

math.MG↗

arXiv2025

The matrix potential game and structures of self-affine sets

We present a new variant of the potential game and show that certain compact subsets of $\mathbb{R}^n$, including a large class of self-affine sets, are winning in our game. We prove that sets with sufficiently strong winning conditions are non-empty, provide a lower bound for their Hausdorff dimension, show that they have good intersection properties, and provide conditions under which, given $M \in \mathbb{N}$, they contain a homothetic copy of every set with at most $M$ elements. The applications of our game to self-affine sets are new and complement the recent work of Yavicoli et al (Math. Z. 2022 and Int. Math. Res. Not. IMRN 2023) for self-similar sets.

math.DS↗