Random continued fractions: Lévy constant and Chernoff-type estimate

Fang, Lulu; Wu, Min; Shieh, Narn-Rueih; Li, Bing

doi:10.1016/j.jmaa.2015.04.013

Cited by 4 publications

(3 citation statements)

References 37 publications

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“…Theorem 3.9 (cf. Theorem 2 in [14] or Theorem B in [49]). For any δ > 0, there exist N > 0, B > 0, α > 0 such that for all n ≥ N ,…”

Section: Gauss Map: Tmentioning

confidence: 99%

Joint normality of representations of numbers: an ergodic approach

Bergelson¹,

Son²

2022

Preprint

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We introduce an ergodic approach to the study of joint normality of representations of numbers.For example, we show that for any integer b ≥ 2 almost every number x ∈ [0, 1) is jointly normal with respect to the b-expansion and continued fraction expansion. This fact is a corollary of the following result which deals with pointwise joint ergodicity:where λ is the Lebesgue measure on [0, 1] and µ G is the Gauss measure on [0, 1] given by µ G (A) =We show that the phenomenon of the pointwise joint ergodicity takes place for a wide variety of numbertheoretical maps of the interval and derive the corresponding corollaries pertaining to joint normality.We also establish the equivalence of various forms of normality and joint normality for representations of numbers, hereby providing a general framework for classical normality results. VITALY BERGELSON AND YOUNGHWAN SONNowadays this result can be easily obtained by means of the ergodic theory. 2 Indeed, consider the map

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“…Theorem 3.9 (cf. Theorem 2 in [14] or Theorem B in [49]). For any δ > 0, there exist N > 0, B > 0, α > 0 such that for all n ≥ N ,…”

Section: Gauss Map: Tmentioning

confidence: 99%

Joint normality of representations of numbers: an ergodic approach

Bergelson¹,

Son²

2022

Preprint

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“…A natural question is arisen: what are the rates of these decreasing probabilities? In fact, Fang et al [8] have considered these decays and showed that the upper bounds of these decays are exponential. In this paper, we not only obtain the upper and lower bounds of these decreasing probabilities, but also give them explicit formulae.…”

Section: Theorem 11 ([14]mentioning

confidence: 99%

The denominators of convergents for continued fractions

Fang¹,

Wu²,

Li³

2016

Preprint

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For any real number x ∈ [0, 1), we denote by qn(x) the denominator of the n-th convergent of the continued fraction expansion of x (n ∈ N). It is well-known that the Lebesgue measure of the set of points x ∈ [0, 1) for which log qn(x)/n deviates away from π 2 /(12 log 2) decays to zero as n tends to infinity. In this paper, we study the rate of this decay by giving an upper bound and a lower bound. What is interesting is that the upper bound is closely related to the Hausdorff dimensions of the level sets for log qn(x)/n. As a consequence, we obtain a large deviation type result for log qn(x)/n, which indicates that the rate of this decay is exponential.

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“…, a m+n in the periodic part of expansion (3). Many results on the Lévy constant have been studied by some authors, see [2,[6][7][8][9]. Some other limit theorems of {q n (x), n ≥ 1} in the continued fraction expansion of real numbers have been extensively studied.…”

Section: Introductionmentioning

confidence: 99%

On the Eventually Periodic Continued β-Fractions and Their Lévy Constants

Xiao

Wang

2022

Mathematics

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In this paper, we consider continued β-fractions with golden ratio base β. We show that if the continued β-fraction expansion of a non-negative real number is eventually periodic, then it is the root of a quadratic irreducible polynomial with the coefficients in Z[β] and we conjecture the converse is false, which is different from Lagrange’s theorem for the regular continued fractions. We prove that the set of Lévy constants of the points with eventually periodic continued β-fraction expansion is dense in [c, +∞), where c=12logβ+2−5β+12.

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Random continued fractions: Lévy constant and Chernoff-type estimate

Cited by 4 publications

References 37 publications

Joint normality of representations of numbers: an ergodic approach

Joint normality of representations of numbers: an ergodic approach

The denominators of convergents for continued fractions

On the Eventually Periodic Continued β-Fractions and Their Lévy Constants

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