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... The upper bound was improved by Erdős and Shallit [2] who leveraged "arithmetic" arguments to combine with the previous "Archimedean" ones. They established P (n) ≪ n 1 3 +ε and also improved the lower bound to lim sup n→∞ P (n)/ log n > 0. These bounds have since remained the state of the art, with the exponent 1/3 representing a natural barrier. ...

... The second bound we present improves this trivial bound, by taking advantage of "arithmetic" properties of the iterative process. It was proven in [2]. We reproduce this proof in our own notation as many of its features make their way into the proof our improvement. ...

... Take some smooth even w so that 2] . Then, we have ...

For a given positive integer $n$, how long can the process $x \mapsto n\text{ }(\text{mod } x)$ last before reaching $0$? We improve Erd\H{o}s and Shallit's upper bound of $O(n^{\frac{1}{3}+\varepsilon})$ to $O(n^{\frac{1}{3}-\frac{2}{177}+\varepsilon})$ for any $\varepsilon > 0$.

... According to [1][2][3][4][5][6][7], it is well-known that each real number 0 < A < 1 can be uniquely representable via several series expansions such as the so-called Sylvester series expansion [1,3,4,6,7] of the form ...

... where a n ∈ N, a 1 ≥ 2, and a n+1 ≥ a n (a n − 1) + 1 for n ≥ 1 and the so-called Engel series expansion [1][2][3][4][5][6][7] of the form ...

... where a n ∈ N and a n+1 ≥ a n (a n + 1) for n ≥ 1 and the alternating-Engel or Pierce series expansion [2,[8][9][10] of the form 2 where a n ∈ N and a n+1 > a n for n ≥ 1. Furthermore, the alternating-Sylvester, respectively, Pierce series expansion for such A is finite if and only if A is rational. ...

In the real number field, there are several unique series expansions for each $A\in (0,1)$. Of interest are the Sylvester and alternating Sylvester series expansions since both expansions are finite if and only if $A$ is rational. We obtain
upper bounds on the length of rational $A\in (0,1)$ and
lower bound on the length of certain classes of rational numbers.
In the power series fields, let $\mathbb{F}_q$ denote the finite field of $q$ elements, let $p(x)$ be an irreducible polynomial in $\mathbb{F}_q[x]$, and let $\mathbb{F}_q((p(x)))$, respectively, $\mathbb{F}_q((1/x))$ be the completions of $\mathbb{F}_q(x)$ with respect to the $p(x)$-adic valuation, respectively, the infinite valuation. It is known that each $A\in \mathbb{F}_q((p(x)))$, respectively, $\mathbb{F}_q((1/x))$, subject to a technical assumption, has a unique Oppenheim series expansion, and such expansion is finite if and only if $A\in\mathbb{F}_q(x)$. Upper bounds on the length of these series expansions for $A\in\mathbb{F}_q(x)$ are also derived.

... Later, Shallit [13] applied this expansion to proposing a very nice method for determining leap years which generalizes those existent in 1994. For more details about the alternating Engel expansion, we refer the reader to [3,4,6,17] and the references therein. Now we turn to introducing the large and moderate deviation principles. ...

... Later, Zhu [18] and Hu [5] studied, respectively, the large deviations and moderate deviations for Engel, Sylvester and Cantor product expansions considered by Erdős et al. [2], which are the classical representations of real numbers in number theory. Although Engel expansion and alternating Engel expansion have some similar properties (see [2,3,12,17]), our Remark 1 still indicates that there is a difference between these two expansions in the context of large deviations. Moreover, we emphasize that the proofs of Zhu [18] and Hu [5] follow from an explicit computation of the Mellin transform of the digit occurring in the Engel expansion and its asymptotic analysis. ...

... In this section, we recall some definitions and several arithmetic and metric properties of the alternating Engel expansion, see [3,4,6,12,13,15,17] for details. We use the notation E(ξ) to denote the expectation of a random variable ξ with respect to the probability measure P. ...

In this paper, we investigate the large and moderate deviation principles for alternating Engel expansions, a classical representation of real numbers in number theory.

... Following Erdos and Shallit [3], we will denote the right-hand side of (14) by the special symbol (a l9 a 2 ,..., a n ,...). If expansion (14) is infinite, a is irrational. ...

... The algorithm presented in this article provides fast best approximations to any irrational of the form -Jr, where r is a positive integer. At the same time, the algorithm provides the necessary background to obtain the Pierce expansion of some quadratic irrationals whose partial quotients, a,, grow as x 3 . The procedure used proves also that the convergents in the Pierce expansions of these irrationals are best approximations of the second kind. ...

In this article two aims are pursued: on the one hand, to present a rapidly converging algorithm for the approximation of square roots; on the other hand and based on the previous algorithm, to find the Pierce expansions of a certain class of quadratic irrationals as an alternative way to the method presented in 1984 by J.O. Shallit; we extend the method to find also the Pierce expansions of quadratic irrationals of the form $2 (p-1) (p - \sqrt{p^2 - 1})$ which are not covered in Shallit's work.

... In 1958, Erdős, Rényi, and Szüsz [7] proved in the last section that FS(a, b) ≤ a and E(a, b) ≤ a. In 1991, Erdős and Shallit [12] obtained an improved bound for E(a, b), namely E(a, b) = O(b 1/3+ϵ ) for all ϵ > 0, and proved that there exists a constant c > 0 such that E(a, b) > c log b infinitely often. For the case of the Fibonacci-Sylvester expansion, Tongron, Kanasri, and Laohakosol [13] improved the upper bound for FS(a, b) mentioned above by showing that ...

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.

... Given a sequence of positive integers (x n ), which is such that x n | x n+1 for all n ≥ 1, the sum of the reciprocals is the Engel series (To ensure convergence it should be assumed that (x n ) is eventually increasing, i.e. for all n there is some n > n with x n > x n .) Every positive real number admits both an Engel expansion, of the form (1.1), and a Pierce expansion (1.2) [6,11], and these S, S are unique: after removing the integer part it is sufficient to consider numbers in the interval (0, 1), and then (x n ) is strictly increasing with x 1 ≥ 2 in (1.1) and x 1 ≥ 1 in (1.2). Although they are not quite so well known, Engel expansions and Pierce expansions have much in common with continued fraction expansions, both in the way that they are determined recursively, and from a metrical point of view; for instance, see [10] for the case of Engel series and [29] for Pierce series. ...

... Given a sequence of positive integers (x n ), which is such that x n | x n+1 for all n, the sum of the reciprocals is the Engel series (To ensure convergence it should be assumed that (x n ) is eventually increasing, i.e. sense that for all n there is some n ′ > n with x n ′ > x n .) Every positive real number admits both an Engel expansion, of the form (1.1), and a Pierce expansion (1.2) [7,12], and these S, S ′ are unique: after removing the integer part it is sufficient to consider numbers in the interval (0, 1), and then (x n ) is strictly increasing with x 1 ≥ 2 in (1.1) and x 1 ≥ 1 in (1.2). Although they are not quite so well known, Engel expansions and Pierce expansions have much in common with continued fraction expansions, both in the way that they are determined recursively, and from a metrical point of view; for instance, see [11] for the case of Engel series and [30] for Pierce series. ...

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 stronger property that $x_n^2$ divides $x_{n+1}$, the continued fraction expansion of the sum is determined explicitly in terms of $z_1=x_1$ and the ratios $z_n=x_n/x_{n-1}^2$ for $n\geq 2$. Here we show that, when this stronger property holds, the same is true for a sum $\sum_{j\geq 1}\epsilon_j/x_j$ with an arbitrary sequence of signs $\epsilon_j=\pm 1$. As an application, we use this result to provide explicit continued fractions for particular families of L\"{u}roth series and alternating L\"{u}roth series defined by nonlinear recurrences of second order. We also calculate exact irrationality exponents for certain families of transcendental numbers defined by such series.

... Maximal lengths and other properties of Pierce and Engel expansions have been studied in [2], [11], [13], and [14]. ...

... These indicate the expansions 1 2 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 32 1 1 2 1 3 2 3 1 4 2 3 2 5 3 6 1 7 4 7 3 5 4 7 2 7 33 1 2 1 2 3 2 5 6 3 6 1 4 7 6 5 6 7 6 9 12 7 2 13 8 13 34 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 1 9 10 10 11 11 12 12 13 35 1 2 3 4 1 2 1 4 5 2 7 8 9 2 3 12 13 14 15 4 3 14 15 16 5 36 1 1 1 1 2 1 2 2 1 2 3 1 3 3 2 2 3 1 4 3 3 5 5 2 6 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 38 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 1 10 11 11 12 12 13 39 1 2 1 2 3 2 3 4 3 4 5 4 1 6 5 6 7 6 7 8 7 10 9 8 11 40 1 1 2 1 1 2 2 1 2 1 3 2 3 3 2 2 5 4 6 1 7 5 7 3 3 41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 42 1 1 1 2 2 1 1 2 2 3 3 2 4 1 3 4 4 3 5 4 1 5 7 4 6 43 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 44 1 1 2 1 2 2 3 2 3 3 1 3 4 4 5 4 6 5 6 5 7 1 8 6 8 45 1 2 1 2 1 2 3 2 1 2 3 2 5 4 1 6 5 2 5 2 3 4 5 4 3 46 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 1 12 13 47 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 48 1 1 1 1 2 1 2 1 2 2 3 1 3 2 2 1 3 2 4 2 3 3 4 1 5 49 1 2 3 4 5 6 1 2 9 10 11 12 13 2 15 16 17 18 13 20 3 22 23 24 25 50 1 1 2 2 1 3 4 4 3 1 4 2 5 7 2 8 9 9 6 2 9 11 8 12 1 b|a 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 26 1 27 14 1 28 7 8 1 29 26 27 28 1 30 7 5 8 9 1 31 26 27 28 29 30 1 32 5 7 3 7 4 5 1 33 14 9 16 15 10 17 18 1 34 13 14 14 15 15 16 16 17 1 35 18 19 4 17 6 23 20 25 26 1 36 4 2 5 7 3 6 6 4 7 8 1 37 26 27 28 29 30 31 32 33 34 35 36 1 38 13 14 14 15 15 16 16 17 17 18 18 19 1 39 2 9 12 11 10 13 14 11 14 15 12 17 18 1 40 5 6 4 7 2 7 4 7 6 3 5 8 6 7 1 41 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 1 42 7 5 2 8 5 7 8 6 9 3 6 10 11 7 12 13 1 43 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 1 44 7 9 7 9 8 9 8 2 9 10 9 11 10 11 10 12 11 12 1 45 8 3 8 11 2 9 10 5 8 5 4 13 12 7 6 15 8 17 18 1 46 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 1 47 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 1 48 4 4 3 6 3 5 2 4 4 6 2 7 5 5 3 6 3 7 4 4 5 6 1 49 26 27 4 29 24 31 32 33 28 5 36 37 38 39 40 35 6 43 44 39 46 41 42 1 50 13 10 14 13 3 8 16 17 17 4 18 17 15 18 4 19 21 22 18 5 23 20 24 25 1 44 ...

In the earlier work, the SEL series expansion for any real number is constructed and characterizations of rational numbers by using such expansion are established. In this paper, we construct the SEL Egyptian fraction expansion for any real number and establish characterizations of rational numbers by using such expansion. These results yield a generalized version of the results for the Fibonacci-Sylvester expansion and the Cohen-Egyptian fraction expansion. A new expansion, called the Lüroth Egyptian fraction expansion is obtained and the relation between the SEL Egyptian fraction expansion and the SEL series expansion is also established.

!!! ЗАУВАЖЕННЯ!!!
Журнал «Науковий часопис НПУ імені М. П. Драгоманова. Серія 1. Фізико-математичні науки» (Trans. Natl. Pedagog. Mykhailo Dragomanov Univ. Ser.1. Phys. Math.) виходить із затримкою у два-чотири роки. При цьому, дати надсилання до редакції статей не проставляються у цьому журналі, статті до друку приймаються до моменту віднесення електронного варіанту відповідного номеру журналу у видавництво (процес видання журналу останнім, зі слів редакторів журналу, займає лише кілька місяців). Даний журнал не входить до науко-метричних баз, примірників усіх випусків цього журналу немає у бібліотеках (у деяких є лише поодинокі випуски), хоча ще на початку 2015 року він вважався фаховим (тобто, у ньому можна було публікувати результати дисертаційних досліджень).
Мій науковий керівник, як один з редакторів цього журналу, на свій розсуд опублікував у ньому шість моїх статей. Друкованих примірників «Наукових часописів НПУ імені М. П. Драгоманова» із моїми статтями мені не видали, мотивуючи тим, що виходить лише кілька примірників «для своїх» із-за якихось фінансових проблем, а для мене є електронні варіанти журналу, що поміщені на сайті Фізико-математичного факультету НПУ імені М. П. Драгоманова (режим доступу: http://fmf.npu.edu.ua/ua/nauka-na-fakulteti/scientific-publications/368-naukchasopys1). Проте, я помітив, що з часом такі електронні варіанти можуть змінюватись: туди докидаються статті інших авторів, змінюються шаблон, сторінки і т. д. Наприклад, з парою різних електронних файлів «Наукового часопису НПУ імені М. П. Драгоманова» за 2012 рік (випуск 13(2)) можна ознайомитись за посиланнями: https://www.researchgate.net/publication/317475200 (перший варіант), https://www.researchgate.net/publication/317475203 (другий варіант).
Даний випуск журналу за 2011 рік вийшов з друку у 2015 році.

We study topological and metric properties of the set
c[[`(o)]1 ,{ Vn } ] = { x:x = ån \frac( - 1)n - 1 g1 (g1 + g2 ) ¼(g1 + g2 + ¼+ gn ),gk Î Vk Ì \mathbbN }c[\bar o^1 ,\{ V_n \} ] = \left\{ {x:x = \sum\limits_n {\frac{{( - 1)^{n - 1} }}{{g_1 (g_1 + g_2 ) \ldots (g_1 + g_2 + \ldots + g_n )}},g_k \in V_k \subset \mathbb{N}} } \right\}
with certain conditions on the sequence of sets {V
n
}. In particular, we establish conditions under which the Lebesgue measure of this set is (a) zero and (b) positive. We compare
the results obtained with the corresponding results for continued fractions and discuss their possible applications to probability
theory.

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