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Diophantine properties of real numbers generated by finite automata. (English) Zbl 1134.11011

Let \(k\geq 2\) be an integer. A sequence \((a_n)_{n\geq 1}\) is said to be \(k\)-automatic if \(a_n\) is a finite state function of the base \(k\)-representation of \(n\). Recently, there has been a lot of activity towards the Diophantine properties of real numbers \(\alpha\in (0,1)\), whose base \(b\) expansion
\[ \alpha=\sum_{n=1}^{\infty} {{a_n}\over {b^n}}, \quad a_n\in \{0,1,\ldots,b-1\},\tag{1} \] (here, \(b\geq 2\) is an integer) has the property that the sequence of digits \({\mathbf a}=(a_n)_{n\geq 1}\) is generated by a \(k\)-automaton. For example, the first author together with Y. Bugeaud proved in [Ann. Math. (2) 165, No. 2, 547–565 (2007; Zbl 1195.11094)], that if \(\alpha\) is irrational, then it is transcendental.
The main result of this paper (Theorem 2.2) is the upper bound \(dk(k^m+1)\) for the approximation exponent of the number (1). Here, \(d\) is the cardinality of the internal alphabet associated to \({\mathbf a}\) and \(m\) is the cardinality of its \(k\)-kernel. For particular numbers \(\alpha\), their method gives a sharper approximation exponent. For example (Theorem 6.1), if \(\zeta_b\) stands for the \(b\)-adic Thue-Morse sequence for which \({\mathbf a}=(1,1,0,1,0,0,1,1,\dots)\) is the binary Thue-Morse sequence, then the approximation exponent of \(\zeta_b\) is at most \(5\). As an immediate consequence of their results, the authors deduce (Theorem 2.1) that the base \(b\)-expansion of a Liouville number cannot be generated by a finite automaton, which was conjectured by J. O. Shallit [“Number Theory and Formal Languages”, in: Emerging applications of number theory, IMA Volumes Berlin: Springer, 547–570 (1999; Zbl 0973.11032)]. They also prove (Theorem 3.3) a particular case of a conjecture attributed to Becker to the effect that irrational automatic real numbers should be \(S\)-numbers in Mahler’s classification. The paper ends with some generalizations of their main results, such as replacing the integer base \(b\geq 2\) by a Pisot or Salem number \(\beta\), etc.
The paper is well-organized and well-written. The proofs are mainly elementary and rely on constructing suitable approximants for \(\alpha\) much in the spirit of the previous work of the first author with Bugeaud, except that several deep theorems are being used such as Cobham’s theorem, a result of Adamczewski, Y. Bugeaud and the reviewer from [C. R. Math. Acad. Sci. Paris 339, No. 1, 11–14 (2004; Zbl 1119.11019)], as well as a Diophantine approximation result of Baker concerning \(U\)-numbers.

MSC:

11B85 Automata sequences
11J82 Measures of irrationality and of transcendence
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