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

Let p be a prime number and let & xi; be an irrational p-adic number. Its multiplicative irrationality exponent & mu;X(& xi;) is the supremum of the real numbers & mu;X for which the inequality lb & xi; alp < labl-& mu;X/2 has infinitely many solutions in nonzero integers a and b. We show that & mu;X(& xi;) can be expressed in terms of a new exponent of approximation attached to a sequence of rational numbers defined in terms of & xi;. We establish that & mu;X(& xi;t,p) = 3, where & xi;t,p is the p-adic number 1 p p2 + p3 p4 + & BULL; & BULL; & BULL;, whose sequence of digits is given by the Thue-Morse sequence over {-1, 1}.