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

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 |bξ − a|p ≤ |ab| /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×(ξt,p) = 3, where ξt,p is the p-adic number 1 − p − p + p − p + . . ., whose sequence of digits is given by the Thue–Morse sequence over {−1, 1}.