Hankel determinants, Padé approximations, and irrationality exponents for p-adic numbers

Abstract

The irrationality exponent of an irrational p-adic number \(\xi \), which measures the approximation rate of \(\xi \) by rational numbers, is in general very difficult to compute explicitly. In this work, we shall show that the irrationality exponents of large classes of automatic p-adic numbers and Mahler p-adic numbers (which are transcendental) are exactly equal to 2. Our classes contain the Thue–Morse–Mahler p-adic numbers, the regular paperfolding p-adic numbers, the Stern p-adic numbers, among others.