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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Perfectly ordered quasicrystals and the Littlewood conjecture
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by Alan Haynes, Henna Koivusalo and James Walton PDF
Trans. Amer. Math. Soc. 370 (2018), 4975-4992 Request permission

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
  • MR Author ID: 707783
  • 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
  • MR Author ID: 1062599
  • 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
  • MR Author ID: 1162597
  • Email: james.j.walton@durham.ac.uk
  • 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
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 4975-4992
  • MSC (2010): Primary 11J13, 52C23
  • DOI: https://doi.org/10.1090/tran/7136
  • MathSciNet review: 3812102