Generalized Thue-Morse sequences of squares
type: Item
instance of: scholarly article
title: Generalized Thue-Morse sequences of squares
zbMATH Open document ID: 1276.11014
author: Michael Drmota
author: Johannes F. Morgenbesser
published in: Israel Journal of Mathematics
publication date: 2012-11-13T00:00:00Z
review text: The Thue-Morse sequence codes the parity of the binary sum of digits function, i.e., its \(n\)th element is the sum of the digits in the binary expansion of \(n\), considered as element of the group \(\mathbb{Z} / 2 \mathbb{Z}\). The authors replace the sum in \(\mathbb{Z} / 2 \mathbb{Z}\) by an arbitrary group action on a compact group \(G\) satisfying the Hausdorff separation axiom; the digits of the \(q\)-ary expansion (with an integer \(q \geq 2\)) are identified with \(e = g_0, g_1, \dots, g_{q-1} \in G\). A generalised Thue-Morse sequence is thus defined by \(T(n) = g_{d_0(n)} g_{d_1(n)} \cdots g_{d_k(n)}\) if \(n = d_0(n) + d_1(n) q + \cdots + d_k(n) q^k\). The main result states that \(T(n^2)\) is \(\nu\)-uniformly distributed in \(G\) with respect to some measure \(\nu\) that is explicitly given, i.e., \(\frac{1}{N} \sum_{0\leq n<N} \delta_{T(n^2)}\) converges weakly to \(\nu\). Setting \(G = \mathbb{Z} / r \mathbb{Z}\), \(g_j = j\), and \(G = \mathbb{R} / \mathbb{Z}\), \(g_j = \alpha j\) with irrational \(\alpha\), respectively, one obtains two main results of \textit{C. Mauduit} and \textit{J. Rivat} [Acta Math. 203, No. 1, 107--148 (2009; Zbl 1278.11076)]. It is also shown that \(T(an+b)\) is uniformly distributed (with respect to Haar measure) in the group generated by \(g_0,g_1,\ldots,g_{q-1}\) if and only if \((a,m) = 1\), where \(m\) is a certain characteristic integer of \(g_0,g_1,\ldots,g_{q-1}\) and \(q\). The Thue-Morse sequence is the prototype of an automatic sequence, and the main motivation of the paper was to generalise the results of Mauduit and Rivat to other automatic sequences. It is a direct consequence of the main result that the frequency of each letter of the sequence \(T(n^2)\) exists when \(T(n)\) is an invertible automatic sequence. Here, invertible means that the sequence is given by an automaton where all transition matrices are invertible and the transition matrix of zero is the identity.
reviewed by: Wolfgang Steiner
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11B85
Mathematics Subject Classification ID: 11K36
Mathematics Subject Classification ID: 11B05
zbMATH DE Number: 6105506
zbMATH Keywords: automatic sequence
zbMATH Keywords: uniform distribution
zbMATH Keywords: letter frequency
zbMATH Keywords: q-multiplicative function
MaRDI profile type: Publication
full work available at URL: s11856-011-0186-2
OpenAlex ID: W2059004789
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DOI: 10.1007/S11856-011-0186-2
description: scientific article; zbMATH DE number 6105506