On the rational approximation to p-adic Thue–Morse numbers

Yann Bugeaud ^[1]
1. [1] University of Strasbourg
  
  University of Strasbourg
  
  Arrondissement de Strasbourg-Ville, Francia
Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 39, Nº 4, 2023, págs. 1349-1371
Idioma: inglés
DOI: 10.4171/RMI/1384
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Resumen
- Let p be a prime number and let be an irrational p-adic number. Its multiplicative irrationality exponent [expression] f is the supremum of the real numbers [expression] for which the inequality [expression has infinitely many solutions in nonzero integers a and b. We show that [expression] can be expressed in terms of a new exponent of approximation attached to a sequence of rational numbers defined in terms of [expression. We establish that [expression / D 3, where [expression];p is the p-adic number 1 p p 2 C p 3 p 4 C [expression], whose sequence of digits is given by the Thue–Morse sequence over [expression].