On the approximation by continued fractions
نویسندگان
چکیده
منابع مشابه
Exponents of Diophantine Approximation and Sturmian Continued Fractions
– 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...
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We establish measures of non-quadraticity and transcendence measures for real numbers whose sequence of partial quotients has sublinear block complexity. The main new ingredient is an improvement of Liouville’s inequality giving a lower bound for the distance between two distinct quadratic real numbers. Furthermore, we discuss the gap between Mahler’s exponent w2 and Koksma’s exponent w ∗ 2 .
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Letting x = [a1(x), a2(x), . . .] denote the continued fraction expansion of an irrational number x ∈ (0, 1), Khinchin proved that Sn(x) = ∑n k=1 ak(x) ∼ 1 log 2 n logn in measure, but not for almost every x. Diamond and Vaaler showed that removing the largest term from Sn(x), the previous asymptotics will hold almost everywhere, showing the crucial influence of the extreme terms of Sn(x) on th...
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ژورنال
عنوان ژورنال: Indagationes Mathematicae (Proceedings)
سال: 1989
ISSN: 1385-7258
DOI: 10.1016/s1385-7258(89)80004-6