Metrical theory for optimal continued fractions
نویسندگان
چکیده
منابع مشابه
Metrical Theory for Optimal Continued Fractions
where Ed= fl, bkEZ2,, k 2 1, and with some constraints on bk and sk. Usually we will assume that x is irrational, and thus that the expansion (1.1) is infinite. A special case of an SRCF is the regular continuedfraction, RCF, which is obtained by taking ck = 1 for every k in (1.1). The aim in introducing the OCF was to optimize two things simultaneously. In the first place one wishes the conver...
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By making fundamental use of the Farey shift map and employing infinite (but σ-finite) measures together with the Chacon-Ornstein ergodic theorem it is possible to find new metrical results for continued fractions. Moreover this offers a unified approach to several existing theorems. The application of ergodic theory to the study of continued fractions began with the Gauss transformation, G: [0...
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Abstract. The Rosen fractions form an infinite family which generalizes the nearestinteger continued fractions. In this paper we introduce a new class of continued fractions related to the Rosen fractions, the α-Rosen fractions. The metrical properties of these α-Rosen fractions are studied. We find planar natural extensions for the associated interval maps, and show that these regions are clos...
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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: Journal of Number Theory
سال: 1990
ISSN: 0022-314X
DOI: 10.1016/0022-314x(90)90135-e