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On the complexity of algebraic numbers. (Sur la complexité des nombres algébriques.) (French) Zbl 1119.11019

Summary: Let \(b> 2\) be an integer. We prove that real numbers whose \(b\)-ary expansion satisfies some given, simple, combinatorial condition are transcendental. This implies that the \(b\)-ary expansion of any algebraic irrational number cannot be generated by a finite automaton.

MSC:

11B85 Automata sequences
11A63 Radix representation; digital problems
11J81 Transcendence (general theory)
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
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