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Perfectly ordered quasicrystals and the Littlewood conjecture


Authors: Alan Haynes, Henna Koivusalo and James Walton
Journal: Trans. Amer. Math. Soc. 370 (2018), 4975-4992
MSC (2010): Primary 11J13, 52C23
DOI: https://doi.org/10.1090/tran/7136
Published electronically: February 8, 2018
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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.


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Additional Information

Alan Haynes
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204
Email: haynes@math.uh.edu

Henna Koivusalo
Affiliation: Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom
Address at time of publication: Department of Mathematics, University of Vienna, Vienna, Austria
Email: henna.koivusalo@univie.ac.at

James Walton
Affiliation: Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom
Address at time of publication: Department of Mathematical Sciences, University of Durham, Durham, United Kingdom
Email: james.j.walton@durham.ac.uk

DOI: https://doi.org/10.1090/tran/7136
Received by editor(s): May 13, 2016
Received by editor(s) in revised form: November 3, 2016
Published electronically: February 8, 2018
Additional Notes: This research was supported by EPSRC grants EP/L001462, EP/J00149X, EP/M023540
Article copyright: © Copyright 2018 American Mathematical Society

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