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Irrationality measures for some automatic real numbers. (English) Zbl 1205.11080

Authors’ summary: This paper is devoted to the rational approximation of automatic real numbers, that is, real numbers whose expansion in an integer base can be generated by a finite automaton. We derive upper bounds for the irrationality exponent of famous automatic real numbers associated with the Thue–Morse, Rudin–Shapiro, paperfolding and Baum–Sweet sequences. These upper bounds arise from the construction of some explicit Padé or Padé type approximants for the generating functions of these sequences. In particular, we prove that the Thue–Morse–Mahler numbers have an irrationality exponent at most equal to 4. We also obtain an explicit description of infinitely many convergents to these numbers.

MSC:

11J04 Homogeneous approximation to one number
11J82 Measures of irrationality and of transcendence
11B85 Automata sequences
68Q45 Formal languages and automata
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References:

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