Perfectly ordered quasicrystals and the Littlewood conjecture

Haynes, A., Koivusalo, H. and Walton, J. (2018) Perfectly ordered quasicrystals and the Littlewood conjecture. Transactions of the American Mathematical Society, 370(7), pp. 4975-4992. (doi: 10.1090/tran/7136)

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.

• Item Type: Articles
• Additional Information: This research was supported by EPSRC grants EP/L001462, EP/J00149X, EP/M023540.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Walton, Dr Jamie
• Authors: Haynes, A., Koivusalo, H., and Walton, J.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Transactions of the American Mathematical Society
• Publisher: American Mathematical Society
• ISSN: 0002-9947
• ISSN (Online): 1088-6850
• Published Online: 08 February 2018
• Copyright Holders: Copyright © 2018 American Mathematical Society
• First Published: First published in Transactions of the American Mathematical Society 370(7): 4975-7992
• Publisher Policy: Reproduced in accordance with the publisher copyright policy

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