2021/10/05 by Bugeaud, Yann
#11J04 #11J61 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2110.01855
Let p be a prime number and ξ an irrational p-adic number. Its multiplicative irrationality exponent μ× (ξ) is the supremum of the real numbers μ× for which the inequality |b ξ- a|p ≤ | a b |^- μ× / 2 has infinitely many solutions in nonzero integers a, b. We show that μ× (ξ) can be expressed in terms of a new exponent of approximation attached to a sequence of rational numbers defined in terms of ξ. We establish that μ× (ξ_\bf t, p) = 3, where ξ_\bf t, p is the p-adic number 1 - p - p2 + p3 - p4 + …, whose sequence of digits is given by the Thue--Morse sequence over \-1, 1\.