Large and Moderate Deviation Principles for Engel Continued Fractions

Abstract

Abstract. Large and moderate deviation principles are proved for Engel continued fractions, a new type of continued fraction expansion with non-decreasing partial quotients in number theory.

A Note on Rényi's ‘Record’ Problem and Engel's Series

“…We use the notation E(ξ) to denote the expectation of a random variable ξ with respect to the probability measure P. In this section, we will give the proofs of Theorems 1.3 and 1.6. In fact, the large deviations for stochastic processes related Markov chain occurring in number theory have been discussed by Zhu [22], Fang [5,6], and Fang and Shang [7]. More precisely, Zhu [22] studied the large deviations for Engel's series whose digit sequence is non-decreasing and forms a time-homogeneous Markov chain with exact one-step transition probabilities.…”

“…More precisely, Zhu [22] studied the large deviations for Engel's series whose digit sequence is non-decreasing and forms a time-homogeneous Markov chain with exact one-step transition probabilities. Fang and Shang [7] considered the large deviations for Engel continued fractions whose digit sequence is also non-decreasing but does not form a Markov chain. Since the digit sequences of Engel's series and Engel continued fractions are non-decreasing, their rate functions have the similar structure in form.…”

“…(see [7,Remark 6] for more details). Replacing C by A in the relevant objects defined as in the proof of Theorem 1.3, we actually deduce that…”

Hausdorff dimension of certain sets arising in Engel continued fractions

“…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]). For the Hausdorff dimension of some sets in ECFs, Zhong and Tang [27] considered a Hirst's problem in the context of ECFs and their results indicate that there is a difference between RCFs and ECFs in this problem.…”

“…) means the Borel σ-algebra on [0,1). We know that {b n } n≥1 does not form a homogeneous Markov chain (see Remark 5 of [7]) but has the following property, which is important in the metric theory of Engel continued fractions (see [4,7,18]). We also emphasize that such a property is not true for regular continued fractions.…”