On Higher Dimensional Arithmetic Progressions in Meyer Sets

Abstract

In this paper we study the existence of higher dimensional arithmetic progressions in Meyer sets. We show that the case when the ratios are linearly dependent over ${\mathbb Z}$ is trivial and focus on arithmetic progressions for which the ratios are linearly independent. Given a Meyer set $\Lambda $ and a fully Euclidean model set with the property that finitely many translates of cover $\Lambda $ , we prove that we can find higher dimensional arithmetic progressions of arbitrary length with k linearly independent ratios in $\Lambda $ if and only if k is at most the rank of the ${\mathbb Z}$ -module generated by . We use this result to characterize the Meyer sets that are subsets of fully Euclidean model sets.

Monochromatic arithmetic progressions in binary Thue–Morse-like words

“…In [5], Avgustinovich and Frid show that any binary word occurs as an arithmetic subsequence of the Thue-Morse sequence (or more generally, a fixed point of any primitive bijective binary constant-length substitution), and go on to investigate properties of the arithmetic complexity of certain words over arbitrary finite alphabets. If we instead consider monochromatic arithmetic subsequences of a given substitutive word and fix the difference of the arithmetic progressions, we note that the length is bounded in the case of the Thue-Morse and more general Thue-Morse-like sequences, as shown, respectively, in [33] and, by the present authors, in [2]; see also [27,28] for results regarding arithmetic progressions in model sets.…”