Beta-expansion and continued fraction expansion of real numbers
نویسندگان
چکیده
منابع مشابه
Ramanujan and the Regular Continued Fraction Expansion of Real Numbers
In some recent papers, the authors considered regular continued fractions of the form [a0; a, · · · , a } {{ } m , a, · · · , a } {{ } m , a, · · · , a } {{ } m , · · · ], where a0 ≥ 0, a ≥ 2 and m ≥ 1 are integers. The limits of such continued fractions, for general a and in the cases m = 1 and m = 2, were given as ratios of certain infinite series. However, these formulae can be derived from ...
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Let x ∈ I be an irrational element and n 1, where I is the unit disc in the field of formal Laurent series F((X−1)), we denote by kn(x) the number of exact partial quotients in continued fraction expansion of x, given by the first n digits in the β-expansion of x, both expansions are based on F((X−1)). We obtain that lim inf n→+∞ kn(x) n = degβ 2Q∗(x) , lim sup n→+∞ kn(x) n = degβ 2Q∗(x) , wher...
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(a general reference is Chapter I of [9]). If ξ is irrational, then, by letting X tend to infinity, this provides infinitely many rational numbers x1/x0 with |ξ − x1/x0| ≤ x 0 . By contrast, an irrational real number ξ is said to be badly approximable if there exists a constant c1 > 0 such that |ξ − p/q| > c1q for each p/q ∈ Q or, equivalently, if ξ has bounded partial quotients in its continue...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 2019
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa170630-27-3