Arithmetic and metric properties of Oppenheim continued fraction expansions

Abstract

We introduce a class of continued fraction expansions called Oppenheim continued fraction (OCF) expansions. Basic properties of these expansions are discussed and metric properties of the digits occurring in the OCF expansions are studied.

Large and Moderate Deviation Principles for Engel Continued Fractions

“…And modified Engel continued fractions is just a small modification of Engel continued fractions. Fan et al [10] have shown that these two continued fractions share the classical limit theorems, such as the law of large numbers, the central limit theorem, the law of the iterated logarithm, and other statical laws. From the fractal points of view, we can obtain that for any α ≥ 0, the set x ∈ (0, 1] : lim n→∞ log e n (x) n = α has full Hausdorff dimension following the idea of [23,24], where {e n (x) : n ≥ 1} is the partial quotient sequence of ECF expansions or modified ECF expansions.…”

“…Engel continued fractions and modified Engel continued fractions (see [11]) are two typical examples of Oppenheim continued fractions introduced by Fan et al [10] in 2007. And modified Engel continued fractions is just a small modification of Engel continued fractions.…”

“…The following proposition, due to Fan, Wang and Wu [10], gives a characterization of all admissible sequences occurring in the ECF expansion.…”

Hausdorff dimension of certain sets arising in Engel continued fractions

“…Moreover, they also showed that the set of real numbers in which such a strong law of large numbers does not hold, has full Hausdorff dimension. Furthermore, Fan et al [4] established a central limit theorem for log b n (x). Following this line of research, Fang et al [7] considered the large and moderate deviation principles for ECF expansions (see also [5,6]).…”

“…In this section, we recall some definitions and several arithmetic properties of Engel continued fractions. We first give an elementary arithmetic property of the Engel continued fraction expansion in representing real numbers, see Hartono et al [12] (see also Fan et al [4]).…”