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Exponents of Diophantine Approximation and Sturmian Continued Fractions

Yann Bugeaud·Michel Laurent
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Abstract

Let x be a real number and let n be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents w_n(x) and w_n^*(x) defined by Mahler and Koksma. We calculate their six values when n=2 and x 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 x by quadratic surds.

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Yann Bugeaud, Michel Laurent. 2004-06-03. Exponents of Diophantine Approximation and Sturmian Continued Fractions. https://arxiv.org/abs/math/0406064

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