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We say that a set\n \n \n $$S\\subseteq \\mathbb {C}$$<\/jats:tex-math>\n \n \n S<\/mml:mi>\n \u2286<\/mml:mo>\n C<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is\n \n \n $$\\lambda $$<\/jats:tex-math>\n \n \u03bb<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -convex<\/jats:italic>\n if, whenever\n a<\/jats:italic>\n and\n b<\/jats:italic>\n are in\n S<\/jats:italic>\n , the point\n \n \n $$(1-\\lambda )a+\\lambda b$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n 1<\/mml:mn>\n -<\/mml:mo>\n \u03bb<\/mml:mi>\n )<\/mml:mo>\n a<\/mml:mi>\n +<\/mml:mo>\n \u03bb<\/mml:mi>\n b<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is also in\n S<\/jats:italic>\n . We investigate the properties of\n \n \n $$\\lambda $$<\/jats:tex-math>\n \n \u03bb<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -convex sets and their (topological) closures, and we prove a number of facts about them. Let\n \n \n $$Q_\\lambda \\subseteq \\mathbb {C}$$<\/jats:tex-math>\n \n \n \n Q<\/mml:mi>\n \u03bb<\/mml:mi>\n <\/mml:msub>\n \u2286<\/mml:mo>\n C<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n be the least\n \n \n $$\\lambda $$<\/jats:tex-math>\n \n \u03bb<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -convex superset of\n \n \n $$\\{0,1\\}$$<\/jats:tex-math>\n \n \n {<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Generalizing results of R. G. E. Pinch, we give a sufficient condition on\n \n \n $$\\lambda $$<\/jats:tex-math>\n \n \u03bb<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for\n \n \n $$Q_\\lambda $$<\/jats:tex-math>\n \n \n Q<\/mml:mi>\n \u03bb<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and some other related\n \n \n $$\\lambda $$<\/jats:tex-math>\n \n \u03bb<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -convex sets to be discrete by introducing the notion of a\n strong PV number<\/jats:italic>\n . These conditions give rise to a number of periodic and aperiodic Meyer sets (often regarded as the mathematical counterpart of \u201cquasicrystals\u201d). This paper is in two parts: Part\u00a0I (Theory, Sections\u00a01\u20136) gives general results about\n \n \n $$\\lambda $$<\/jats:tex-math>\n \n \u03bb<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -convex sets and explores the connections between\n \n \n $$\\lambda $$<\/jats:tex-math>\n \n \u03bb<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -convex sets and quasicrystals; Part\u00a0II (Applications, Sections\u00a07\u201311) applies the results of Part\u00a0I to a number of special cases. In Part\u00a0II, we also display several aperiodic\n \n \n $$\\lambda $$<\/jats:tex-math>\n \n \u03bb<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -convex sets, including several with dihedral symmetry. Section\u00a011 in Part\u00a0II 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 Mas\u00e1kov\u00e1, Patera, & Pelantov\u00e1. An extended paper that includes the results given here among many others is available online.\n <\/jats:p>","DOI":"10.1007\/s00454-025-00816-4","type":"journal-article","created":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T13:17:54Z","timestamp":1781183874000},"page":"1-74","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Fixed-Parameter Extrapolation and Aperiodic Order"],"prefix":"10.1007","volume":"76","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6028-9323","authenticated-orcid":false,"given":"Stephen","family":"Fenner","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Frederic","family":"Green","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Steven","family":"Homer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,6,11]]},"reference":[{"key":"816_CR1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511546563","volume-title":"Automatic Sequences: Theory, Applications","author":"J-P Allouche","year":"2003","unstructured":"Allouche, J.-P., Shallit, J.: Automatic Sequences: Theory, Applications. 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