Exponents of Diophantine Approximation and Sturmian Continued Fractions

Abstract

Let ξ be a real number and let n be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents w n (ξ) and w n * (ξ) defined by Mahler and Koksma. We calculate their six values when n=2 and ξ is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we obtain the exact exponent of approximation to such a continued fraction ξ by quadratic surds.