Mathematics
Document Type
Article
Abstract
Fix any λ∈C. We say that a set S⊆C is λ-convex if, whenever a and b are in S, the point (1-λ)a+λb is also in S. We investigate the properties of λ-convex sets and their (topological) closures, and we prove a number of facts about them. Let Qλ⊆C be the least λ-convex superset of {0,1}. Generalizing results of R. G. E. Pinch, we give a sufficient condition on λ for Qλ and some other related λ-convex sets to be discrete by introducing the notion of a strong PV number. These conditions give rise to a number of periodic and aperiodic Meyer sets (often regarded as the mathematical counterpart of “quasicrystals”). This paper is in two parts: Part I (Theory, Sections 1–6) gives general results about λ-convex sets and explores the connections between λ-convex sets and quasicrystals; Part II (Applications, Sections 7–11) applies the results of Part I to a number of special cases. In Part II, we also display several aperiodic λ-convex sets, including several with dihedral symmetry. Section 11 in Part II contains conjectures, open problems, and other suggestions for further research. Our work combines elementary concepts and techniques from algebra and plane geometry. Our results extend and generalize previous work of Pinch, Berman & Moody, and Masáková, Patera, & Pelantová. An extended paper that includes the results given here among many others is available online. © The Author(s) 2026.
Publication Title
Discrete and Computational Geometry
Publication Date
7-2026
Volume
76
Issue
1
First Page
1
Last Page
74
ISSN
0179-5376
DOI
10.1007/s00454-025-00816-4
Keywords
a-convex, Aperiodic order, cut-and-project scheme, Discrete geometry, Meyer set, Quasicrystal
Repository Citation
Fenner, Stephen; Green, Frederic; and Homer, Steven, "Fixed-Parameter Extrapolation and Aperiodic Order" (2026). Mathematics. 55.
https://commons.clarku.edu/mathematics/55
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Copyright Conditions
Fenner, S., Green, F., & Homer, S. (2026). Fixed-Parameter Extrapolation and Aperiodic Order. Discrete & Computational Geometry, 1-74.
