Perfectly ordered quasicrystals and the Littlewood conjecture

TL;DR

This paper explores the concept of linear repetitivity in cut and project sets as models for quasicrystals, linking their existence to the unresolved Littlewood conjecture in Diophantine approximation.

Contribution

It extends the classical definition of linear repetitivity and constructs a large class of such sets, revealing their connection to a major open problem in number theory.

Findings

Uncountable collection of linearly repetitive sets with large Hausdorff dimension.

Existence of these sets is equivalent to the negation of the Littlewood conjecture.

Provides new insights into the structure of quasicrystal models.

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.