Perfectly ordered quasicrystals and the Littlewood conjecture

Alan Haynes, Henna L L Koivusalo, James Walton

Research output: Contribution to journalArticle (Academic Journal)peer-review

1 Citation (Scopus)
17 Downloads (Pure)

Abstract

Linearly repetitive cut and project sets are mathematical models for perfectly ordered quasicrystals. In a previous paper we presented a characterization of linearly repetitive cut and project sets. In this paper we extend the classical definition of linear repetitivity to try to discover whether or not there is a natural class of cut and project sets which are models for quasicrystals which are better than `perfectly ordered'. In the positive direction, we demonstrate an uncountable collection of such sets (in fact, a collection with large Hausdorff dimension) for every choice of dimension of the physical space. On the other hand, we show that, for many natural versions of the problems under consideration, the existence of these sets turns out to be equivalent to the negation of a well-known open problem in Diophantine approximation, the Littlewood conjecture.

Original languageEnglish
Pages (from-to)4975-4992
Number of pages18
JournalTransactions of the American Mathematical Society
Volume370
DOIs
Publication statusPublished - 8 Feb 2018

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