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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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