Perfectly ordered quasicrystals and the Littlewood conjecture

Haynes, Alan; Koivusalo, Henna; Walton, James

doi:10.1090/tran/7136

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
MathSciNet review: 3812102
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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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