On $p$-adic $T$-numbers

Abstract

Denote by w_n* and w_n the exponents of Diophantine approximation defined in Mahler’s and Koksma’s classifications of transcendental numbers, respectively. We prove that there are p-adic T-numbers x such that w_n(x) − w_n*(x) is any number chosen in the segment [0, (n −1)/n] for all positive integers n and for w_n(x) large enough. Thus we improve Schlickewei’s result that p-adic T-numbers do exist.