Engel Series Research Articles

MULTIFRACTAL ANALYSIS OF THE DIVERGENCE POINTS ASSOCIATED WITH THE GROWTH OF DIGITS IN ENGEL EXPANSIONS

In this paper, we are concerned with the multifractal analysis of the divergence points in Engel expansions. Let [Formula: see text] be an irrational number with Engel expansion [Formula: see text]. For any [Formula: see text], let [Formula: see text] We prove that the Hausdorff dimension of [Formula: see text] is [Formula: see text] when [Formula: see text], and it is zero when [Formula: see text]. This indicates that the Hausdorff dimension of [Formula: see text] is independent of [Formula: see text]. A very different phenomenon is shown for the gap of consecutive digits. For any irrational number [Formula: see text] and [Formula: see text], let [Formula: see text] with [Formula: see text]. We derive that, for any [Formula: see text], the set [Formula: see text] has Hausdorff dimension [Formula: see text].

On the exact rate of convergence of digits in Engel expansions

Let ϕ:N→R+ be a function satisfying ϕ(n)→∞ as n→∞. Denote by 〈d1(x),d2(x),⋯,dn(x),⋯〉 the Engel expansion of an irrational number x∈(0,1). We study the Baire classification and Hausdorff dimension ofDϕ(α,β):={x∈(0,1)﹨Q:liminfn→∞log⁡dn(x)−nϕ(n)=α,limsupn→∞log⁡dn(x)−nϕ(n)=β} for α,β∈[−∞,∞] with α≤β. Under the condition that ϕ(n+1)−ϕ(n)→0 as n→∞, we show that the set Dϕ(α,β) is residual if and only if α=−∞ and β=∞. Under the conditions that ϕ is increasing and ϕ(n+1)−ϕ(n)→0 as n→∞, we prove that Dϕ(α,β) has full Hausdorff dimension for any −∞≤α≤β≤∞, which improves the result of Liu and Wu (2003). We also do some similar analyses for the gap of consecutive digits.

Localized growth speed of the digits in Engel expansions

We introduce a localized version of a Galambos's question about the growth speed of the digits in Engel expansion, namely the set{x∈(0,1]:limn→∞⁡1nlog⁡dn(x)=α(x)}, where α:[0,1]→[0,∞] is a nonnegative continuous function and dn(x) denotes the nth digits in the Engel expansion of x. The Hausdorff dimension is shown to be irrelevant of the function α(x). As applications, this answers Galambos's question and strengthens some results by Liu & Wu.

One class of continuous functionswith complicated local properties related\newline to Engel series

In the paper, we construct and study the class of continuous on $[0, 1]$functions with continuum set of peculiarities(singular, nowhere monotonic, and non-differentiable functionsare among them).The representative of this class is the function $y = f(x)$defined by the Engel representation of argument:\begin{align*}x = \sum_{n=1}^\infty\frac{1}{(2+g_1)(2+g_1+g_2)\ldots(2+g_1+g_2+\ldots+g_n)}=: \Delta^E_{g_1g_2\ldots g_n\ldots},\end{align*}where $g_n = g_n(x) \in \{ 0, 1, 2, \ldots \}$, and convergent real series\[\sum_{n=0}^\infty u_n = u_0 + u_1 + \ldots + u_n + r_n = 1,\qquad\lvert u_n \rvert < 1,\ 0 < r_n < 1,\]by the following equality\[f(\Delta^E_{g_1(x)g_2(x)\ldots g_n(x)\ldots})= r_{g_1(x)} + \sum_{k=2}^\infty\biggl( r_{g_k(x)} \prod_{i=1}^{k-1} u_{g_i(x)} \biggr).\]We study local and global properties of function $f$:structural, extremal, differential, integral, and fractal properties.

On the SEL Egyptian fraction expansion for real numbers

&lt;abstract&gt;&lt;p&gt;In the authors' earlier work, the SEL Egyptian fraction expansion for any real number was constructed and characterizations of rational numbers by using such expansion were established. These results yield a generalized version of the results for the Fibonacci-Sylvester and the Engel series expansions. Under a certain condition, one of such characterizations also states that the SEL Egyptian fraction expansion is finite if and only if it represents a rational number. In this paper, we obtain an upper bound for the length of the SEL Egyptian fraction expansion for rational numbers, and the exact length of this expansion for a certain class of rational numbers is verified. Using such expansion, not only is a large class of transcendental numbers constructed, but also an explicit bijection between the set of positive real numbers and the set of sequences of nonnegative integers is established.&lt;/p&gt;&lt;/abstract&gt;

On the exponent of convergence of the digit sequence of Engel series

For x∈(0,1), let 〈d1(x),d2(x),d3(x),⋯〉 be the Engel series expansion of x. Denote by λ(x) the exponent of convergence of the sequence {dn(x)}, namelyλ(x)=inf⁡{s≥0:∑n≥1dn−s(x)<∞}. It follows from Erdős, Rényi and Szüsz (1958) that λ(x)=0 for Lebesgue almost all x∈(0,1). This paper is concerned with the topological and fractal properties of the level set {x∈(0,1):λ(x)=α} for α∈[0,∞]. For the topological properties, it is proved that each level set is uncountable and dense in (0,1), and the level set is of first category for α∈[0,∞) but residual for α=∞. For the fractal properties, we prove that the Hausdorff dimensions of the level sets are as follows:dimH⁡{x∈(0,1):λ(x)=α}={1−α,0≤α≤1;0,1<α≤∞.

THE RELATIVE CONVERGENCE SPEED FOR ENGEL EXPANSIONS AND HAUSDORFF DIMENSION

In this paper, we investigate how many real numbers can be well approximated by their convergents in the Engel expansions. Furthermore, the relative growth rate of convergence speed of convergents in the Engel expansion of an irrational number is studied to the rate of growth of its digits. The Hausdorff dimension of exceptional sets of points with a given relative growth rate is established.

Continued fractions for strong Engel series and Lüroth series with signs

An Engel series is a sum of reciprocals $\sum _{j\geq 1} 1/x_j$ of a non-decreasing sequence of positive integers $x_n$ with the property that $x_n$ divides $x_{n+1}$ for all $n\geq 1$. In previous work, we have shown that for any Engel series with the st

ON SOME EXCEPTIONAL SETS IN ENGEL EXPANSIONS AND HAUSDORFF DIMENSIONS

For any [Formula: see text], let the infinite series [Formula: see text] be the Engel expansion of [Formula: see text]. Suppose [Formula: see text] is a strictly increasing function with [Formula: see text] and let [Formula: see text], [Formula: see text] and [Formula: see text] be defined as the sets of numbers [Formula: see text] for which the limit, upper limit and lower limit of [Formula: see text] is equal to [Formula: see text]. In this paper, we qualify the size of the set [Formula: see text], [Formula: see text] and [Formula: see text] in the sense of Hausdorff dimension and show that these three dimensions can be different.

On the growth speed of digits in Engel expansions

Let ϕ:N→R+ be a non-decreasing function such that ϕ(n)/log⁡n →∞ as n→∞. In this paper, we completely determine the Hausdorff dimension of the setEϕ={x∈(0,1]:limn→∞⁡log⁡dn(x)ϕ(n)=1}, where dn(x) is the digit of the Engel expansion of x. It significantly generalises the existing results on the Hausdorff dimension of Eϕ. Besides, some analogous results for gaps and ratios of consecutive digits are also provided.

Numerical Characteristics of a Random Variable Related to the Engel Expansions of Real Numbers

It is known that any number x ∈ (0; 1] ≡ Ω has a unique Engel expansion $$ x=\sum \limits_{n=1}^{\infty}\frac{1}{\left({p}_1(x)+1\right)\dots \left({p}_n(x)+1\right)}, $$ where pn(x) ∈ ℕ, pn+1(x) ≥ pn(x) for all n ∈ ℕ. This means that pn(x) is a well-defined measurable function on the probability space (Ω, ℱ, λ), where ℱ is the σ-algebra of Lebesgue-measurable subsets of Ω and λ is the Lebesgue measure. The main subject of our research is a function $$ \psi (x)=\sum \limits_{n=1}^{\infty}\frac{1}{p_n(x)+1}, $$ defined on Ω* ⊂ Ω, where Ω* is the set of convergence of the series $$ {\sum}_{n=1}^{\infty}\frac{1}{p_n(x)+1} $$ . It is proved that the function ψ is defined (takes finite values) a.e. on (0; 1]. Moreover, it is shown that ψ is a random variable on the probability space (Ω*, ℱ∗, λ), where ℱ∗ is the σ-algebra of Lebesgue-measurable subsets of Ω*. We determine the mathematical expectation and variance of the function ψ. Furthermore, we consider random variables ψk as a generalization of the function ψ and find their mathematical expectations Mψk.

SLOW GROWTH RATE OF THE DIGITS IN ENGEL EXPANSIONS

We are concerned with the Hausdorff dimension of the set [Formula: see text] where [Formula: see text] is the digit of the Engel expansion of [Formula: see text] and [Formula: see text] is a function such that [Formula: see text] as [Formula: see text]. The Hausdorff dimension of [Formula: see text] is studied by Lü and Liu [Hausdorff dimensions of some exceptional sets in Engel expansions, J. Number Theory 185 (2018) 490–498] under the condition that [Formula: see text] grows to infinity. The aim of this paper is to determine the Hausdorff dimension of [Formula: see text] when [Formula: see text] slowly increases to infinity, such as in logarithmic functions and power functions with small exponents. We also provide a detailed analysis of the gaps between consecutive digits. This includes the central limit theorem and law of the iterated logarithm for [Formula: see text] and the Hausdorff dimension of the set [Formula: see text] where [Formula: see text] with the convention [Formula: see text].

Continued fractions and irrationality exponents for modified Engel and Pierce series

An Engel series is a sum of reciprocals of a non-decreasing sequence (x_n) of positive integers, which is such that each term is divisible by the previous one, and a Pierce series is an alternating sum of the reciprocals of a sequence with the same property. Given an arbitrary rational number, we show that there is a family of Engel series which when added to it produces a transcendental number alpha whose continued fraction expansion is determined explicitly by the corresponding sequence (x_n), where the latter is generated by a certain nonlinear recurrence of second order. We also present an analogous result for a rational number with a Pierce series added to or subtracted from it. In both situations (a rational number combined with either an Engel or a Pierce series), the irrationality exponent is bounded below by (3+sqrt{5})/2, and we further identify infinite families of transcendental numbers alpha whose irrationality exponent can be computed precisely. In addition, we construct the continued fraction expansion for an arbitrary rational number added to an Engel series with the stronger property that x_j^2 divides x_{j+1} for all j.

Hausdorff dimension of certain sets arising in Engel expansions

The present paper is concerned with the Hausdorff dimension of certain sets arising in Engel expansions. In particular, the Hausdorff dimension of the set is completely determined, where An(x) can stand for the digit, gap and ratio between two consecutive digits in the Engel expansion of x and ϕ is a positive function defined on natural numbers. These results significantly extend the existing results of Galambos’ open problems on the Hausdorff dimension of sets related to the growth rate of digits.

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

AbstractIn 1973, Williams [D. Williams, On Rényi's ‘record’ problem and Engel's series, Bull. London Math. Soc.5 (1973), 235–237] introduced two interesting discrete Markov processes, namely C-processes and A-processes, which are related to record times in statistics and Engel's series in number theory respectively. Moreover, he showed that these two processes share the same classical limit theorems, such as the law of large numbers, central limit theorem and law of the iterated logarithm. In this paper, we consider the large deviations for these two Markov processes, which indicate that there is a difference between C-processes and A-processes in the context of large deviations.

On singularity of distribution of random variables with independent symbols of Oppenheim expansions

The paper is devoted to the restricted Oppenheim expansion of real numbers ($ROE$), which includes already known Engel, Sylvester and Luroth expansions as partial cases. We find conditions under which for almost all (with respect to Lebesgue measure) real numbers from the unit interval their $ROE$-expansion contain arbitrary digit $i$ only finitely many times. Main results of the paper state the singularity (w.r.t. the Lebesgue measure) of the distribution of a random variable with i.i.d. increments of symbols of the restricted Oppenheim expansion. General non-i.i.d. case is also studied and sufficient conditions for the singularity of the corresponding probability distributions are found.

Hausdorff dimensions of some exceptional sets in Engel expansions

Given any real number x∈(0,1], denote its Engel expansion by ∑n=1∞1d1(x)⋯dn(x), where {dj(x),j≥1} is a sequence of positive integers satisfying d1(x)≥2 and dj+1(x)≥dj(x) (j≥1). Suppose ϕ:N→R+ is a function satisfying ϕ(n+1)−ϕ(n)→∞ as n→∞. In this paper, we consider the setE(ϕ)={x∈(0,1]:limn→∞⁡log⁡dn(x)ϕ(n)=1}, and we quantify the size of E(ϕ) in the sense of Hausdorff dimension. As applications, we get the Hausdorff dimensions of the sets {x∈(0,1]:limn→∞⁡log⁡dn(x)nβ=γ} and {x∈(0,1]:limn→∞⁡log⁡dn(x)τn=η}, where β>1,γ>0 and τ>1,η>0.

How the dimension of some GCFϵ sets change with proper choice of the parameter function ϵ(k)

For a parameter function ϵ(k) satisfying the condition ϵ(k)+k+1>0, let x=[k1(x),k2(x),⋯]ϵ denote the GCFϵ expansion of x. In this paper, we consider the fractional set asEϵ(a,b)={x∈(0,1):kn(x)≥abnfor infinitely manyn∈N} and obtain that:dimH⁡Eϵ(a,b)={11+b,when ϵ(k)=−k;1b,when −kρ≤ϵ(k)≤k and ρ<1;1b−β+1,when ϵ(k)∼kβ and b≥β≥1;1,when ϵ(k)∼kβ and b≤β, where real a,b>1, and dimH denotes the Hausdorff dimension.It is interesting that when we choose ϵ(k)=−k, Eϵ(a,b) has the same size with the similar set of regular continued fractions; and when −kρ≤ϵ(k)≤k and ρ<1, Eϵ(a,b) has the same size with the similar set of Engel series.

Large and moderate deviations for modified Engel series

ABSTRACTThe modified Engel series of real numbers introduced by Rényi (1962) is a simple modification of Engel series, and they have the same classical limit theorems, such as the law of large numbers, central limit theorem, and law of the iterated logarithm. In this paper, we studied the large and moderate deviations for modified Engel series, which indicate that the large deviations for modified Engel series and Engel series are different.

On the continued fraction expansion of certain Engel series

TextAn Engel series is a sum of the reciprocals of an increasing sequence of positive integers, which is such that each term is divisible by the previous one. Here we consider a particular class of Engel series, for which each term of the sequence is divisible by the square of the preceding one, and find an explicit expression for the continued fraction expansion of the sum of a generic series of this kind. As a special case, this includes certain series whose continued fraction expansion was found by Shallit. A family of examples generated by nonlinear recurrences with the Laurent property is considered in detail, along with some associated transcendental numbers. VideoFor a video summary of this paper, please visit https://youtu.be/T7OL_rKI8Z8.