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Exponents of Diophantine approximation and Sturmian continued fractions. (English) Zbl 1155.11333
Summary: Let $$\xi$$ be a real number and let $$n$$ be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents $$w_n(\xi)$$ and $$w_n^*(\xi)$$ defined by Mahler and Koksma. We calculate their six values when $$n=2$$ and $$\xi$$ 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 $$\xi$$ by quadratic surds.

##### MSC:
 11J13 Simultaneous homogeneous approximation, linear forms 11J82 Measures of irrationality and of transcendence 11J70 Continued fractions and generalizations
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