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1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is, (1) $x=1/d₁\left(x\right)+1/\left(d₁\left(x\right)d₂\left(x\right)\right)+...+1/(d₁\left(x\right)d₂\left(x\right)...d_n\left(x\right))+...$, where $d_j\left(x\right),j\ge 1$ is a sequence of positive integers satisfying d₁(x) ≥ 2 and $d_{j+1}\left(x\right)\ge d_j\left(x\right)$ for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1], (2) $li{m}_{n\to \infty }{d}_n^{1/n}\left(x\right)=e.$He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture: Theorem. $di{m}_{H}x\in (0,1]:\left(2\right)fails=1$. We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and $di{m}_H$ to denote the Hausdorff...

For any $x\in (0,1]$, let $$x=\frac{1}{d_1}+\frac{1}{d_1(d_1-1)d_2}+\cdots +\frac{1}{d_1(d_1-1)\cdots d_{n-1}(d_{n-1}-1)d_n}+\cdots$$ be its Lüroth expansion. Denote by $P_n\left(x\right)/Q_n\left(x\right)$ the partial sum of the first $n$ terms in the above series and call it the $n$th convergent of $x$ in the Lüroth expansion. This paper is concerned with the efficiency of approximating real numbers by their convergents ${\{P_n\left(x\right)/Q_n\left(x\right)\}}_{n\ge 1}$ in the Lüroth expansion. It is shown that almost no points can have convergents as the optimal approximation for infinitely many times in the Lüroth expansion. Consequently, Hausdorff dimension is introduced to quantify the set of...

For x ∈ (0,1), the univoque set for x, denoted (x), is defined to be the set of β ∈ (1,2) such that x has only one representation of the form x = x₁/β + x₂/β² + ⋯ with $x_i\in 0,1$. We prove that for any x ∈ (0,1), (x) contains a sequence ${{\beta }_k}_{k\ge 1}$ increasing to 2. Moreover, (x) is a Lebesgue null set of Hausdorff dimension 1; both (x) and its closure $\overline{\left(x\right)}$ are nowhere dense.

Let $S$ be a non-empty subset of positive integers. A partition of a positive integer $n$ into $S$ is a finite nondecreasing sequence of positive integers $a_1,a_2,\cdots ,a_r$ in $S$ with repetitions allowed such that ${\sum }_{i=1}^r{a}_i=n$. Here we apply Pólya’s enumeration theorem to find the number $\P (n;S)$ of partitions of $n$ into $S$, and the number $\mathrm{D}P(n;S)$ of distinct partitions of $n$ into $S$. We also present recursive formulas for computing $\P (n;S)$ and $\mathrm{D}P(n;S)$.

Let ${fₙ}_{n\ge 1}$ be an infinite iterated function system on [0,1] satisfying the open set condition with the open set (0,1) and let Λ be its attractor. Then to any x ∈ Λ (except at most countably many points) corresponds a unique sequence ${aₙ\left(x\right)}_{n\ge 1}$ of integers, called the digit sequence of x, such that $x=li{m}_{n\to \infty }{f}_{a₁\left(x\right)}\circ \cdots \circ f_{aₙ\left(x\right)}\left(1\right)$. We investigate the growth speed of the digits in a general infinite iterated function system. More precisely, we determine the dimension of the set $x\in \Lambda :aₙ\left(x\right)\in B\left(\forall n\ge 1\right),li{m}_{n\to \infty }aₙ\left(x\right)=\infty$ for any infinite subset B ⊂ ℕ, a question posed by Hirst for continued...

Let $\varphi$ be a substitution over a 2-letter alphabet, say $\{a,b\}$. If $\varphi \left(a\right)$ and $\varphi \left(b\right)$ begin with $a$ and $b$ respectively, $\varphi$ has two fixed points beginning with $a$ and $b$ respectively. We characterize substitutions with two cofinal fixed points (i.e., which differ only by prefixes). The proof is a combinatorial one, based on the study of repetitions of words in the fixed points.

Let $E$ be a self-similar set with similarities ratio $r_j(0\le j\le m-1)$ and Hausdorff dimension $s$, let $\overrightarrow{p}(p_0,p_1)...p_{m-1}$ be a probability vector. The Besicovitch-type subset of $E$ is defined as $$E\left(\overrightarrow{p}\right)=\left\{x\in E:\underset{n\to \infty }{lim}\frac{1}{n}\sum _{k=1}^n{\chi }_j\left(x_k\right)=p_j,0\le j\le m-1\right\},$$ where ${\chi }_j$ is the indicator function of the set $\left\{j\right\}$. Let $\alpha ={dim}_H\left(E\left(\overrightarrow{p}\right)\right)={dim}_P\left(E\left(\overrightarrow{p}\right)\right)=\frac{{\sum }_{j=0}^{m-1}{p}_{j}log{p}_j}{{\sum }_{j=0}^{m-1}{p}_{i}log{r}_j}$ and $g$ be a gauge function, then we prove in this paper:(i) If $\overrightarrow{p}=(r_0^s,r_1^s,\cdots ,r_{m-1}^s)$, then $${\mathscr{H}}^s\left(E\left(\overrightarrow{p}\right)\right)={\mathscr{H}}^s\left(E\right),{𝒫}^s\left(E\left(\overrightarrow{p}\right)\right)={𝒫}^s\left(E\right),$$ moreover both of ${\mathscr{H}}^s\left(E\right)$ and ${𝒫}^s\left(E\right)$ are finite positive;(ii) If $\overrightarrow{p}$ is a positive probability vector other than $(r_0^s,r_1^s,\cdots ,r_{m-1}^s)$, then the gauge functions can be partitioned as follows ...

Fault tolerant control for uncertain systems with time varying state-delay is studied in this paper. Based on sliding mode controller design, a fault tolerant control method is proposed. By means of the feasibility of some linear matrix inequalities (LMIs), delay dependent sufficient condition is derived for the existence of a linear sliding surface which guarantees quadratic stability of the reduced-order equivalent system restricted to the sliding surface. A reaching motion controller, which can...