Generalized Thue-Morse sequences of squares
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.
scientific article; zbMATH DE number 6105506