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On variations of the Liouville constant which are also Liouville numbers. (English) Zbl 1341.11039

Summary: Let \(\ell\) be the Liouville’s constant, defined as a decimal with a 1 in each decimal place corresponding to \(n!\) and 0 otherwise. This number is a classical example of a Liouville number. In this note, we give an optimal condition on the number of replacements of 0’s by 1’s between two consecutive 1’s in the decimal expansion of \(\ell\) in order to ensure that this new number is still a Liouville number.

MSC:

11J81 Transcendence (general theory)
11K60 Diophantine approximation in probabilistic number theory
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References:

[1] J. Liouville, Nouvelle démonstration d’un théorème sur les irrationnelles algébriques, inséré dans le Compte rendu de la dernière séance, C. R. Acad. Sci. Paris 18 (1844), 910-911.
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