Page 1 Next

Displaying 1 – 20 of 173

Showing per page

Integers

A fibered system associated with the prime number sequence. (Sur un système fibré lié à la suite des nombres premiers.)

Experimental Mathematics

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

A note on the approximation by continued fractions under an extra condition.

The New York Journal of Mathematics [electronic only]

Acta Arithmetica

A Note on the Voronoi Summation Formula.

Monatshefte für Mathematik

A problem of Galambos on Engel expansions

Acta Arithmetica

1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is, (1) $x=1/d₁\left(x\right)+1/\left(d₁\left(x\right)d₂\left(x\right)\right)+...+1/\left(d₁\left(x\right)d₂\left(x\right)...{d}_{n}\left(x\right)\right)+...$, 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. $di{m}_{H}x\in \left(0,1\right]:\left(2\right)fails=1$. We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and $di{m}_{H}$ to denote the Hausdorff...

A theorem of H. Steinhaus and $\left(R\right)$-dense sets of positive integers

Czechoslovak Mathematical Journal

A two-dimensional univoque set

Fundamenta Mathematicae

Let J ⊂ ℝ² be the set of couples (x,q) with q > 1 such that x has at least one representation of the form $x={\sum }_{i=1}^{\infty }{c}_{i}{q}^{-i}$ with integer coefficients ${c}_{i}$ satisfying $0\le {c}_{i}, i ≥ 1. In this case we say that $\left({c}_{i}\right)=c₁c₂...$ is an expansion of x in base q. Let U be the set of couples (x,q) ∈ J such that x has exactly one expansion in base q. In this paper we deduce some topological and combinatorial properties of the set U. We characterize the closure of U, and we determine its Hausdorff dimension. For (x,q) ∈ J, we also prove new properties...

Absolut konvergente Reihen und das Hausdorffsche Mass

Czechoslovak Mathematical Journal

Séminaire Delange-Pisot-Poitou. Théorie des nombres

Almost periodic functions and uniform distribution mod 1.

Journal für die reine und angewandte Mathematik

Application de la théorie des algèbres de mesures à l'étude des mesures spectrales.

Seminaire de Théorie des Nombres de Bordeaux

Applications de la notion d'entropie au développement d'un nombre réel dans une base de Pisot

Annales de l'institut Fourier

Soit $\theta$ un nombre de Pisot de degré $s$ ; nous avons montré précédemment que l’endomorphisme du tore ${\mathbf{T}}^{s}$ dont $\theta$ est valeur propre est facteur du $\theta$-shift bilatéral par une application continue ${q}_{s}$ ; nous prouvons ici (théorème 1) que l’application ${q}_{s}$ conserve l’entropie de toute mesure invariante sur le $\theta$-shift. Ceci permet de définir l’entropie d’un nombre dans la base $\theta$ et d’en étudier la stabilité. Nous généralisons également des résultats de Kamae, Rauzy et Bernay.

Asymptotic densities of sets of positive integers

Mathematica Slovaca

Aus der Theorie der zahlentheoretischen Funktionen.

Jahresbericht der Deutschen Mathematiker-Vereinigung

Besicovitch subsets of self-similar sets

Annales de l’institut Fourier

Let $E$ be a self-similar set with similarities ratio ${r}_{j}\left(0\le j\le m-1\right)$ and Hausdorff dimension $s$, let $\stackrel{\to }{p}\left({p}_{0},{p}_{1}\right)...{p}_{m-1}$ be a probability vector. The Besicovitch-type subset of $E$ is defined as$E\left(\stackrel{\to }{p}\right)=\left\{x\in E:\underset{n\to \infty }{lim}\frac{1}{n}\sum _{k=1}^{n}{\chi }_{j}\left({x}_{k}\right)={p}_{j},\phantom{\rule{1.0em}{0ex}}0\le j\le m-1\right\},$where ${\chi }_{j}$ is the indicator function of the set $\left\{j\right\}$. Let $\alpha ={dim}_{H}\left(E\left(\stackrel{\to }{p}\right)\right)={dim}_{P}\left(E\left(\stackrel{\to }{p}\right)\right)=\frac{{\sum }_{j=0}^{m-1}{p}_{j}log{p}_{j}}{{\sum }_{j=0}^{m-1}{p}_{i}log{r}_{j}}$ and $g$ be a gauge function, then we prove in this paper:(i) If $\stackrel{\to }{p}=\left({r}_{0}^{s},{r}_{1}^{s},\cdots ,{r}_{m-1}^{s}\right)$, then${ℋ}^{s}\left(E\left(\stackrel{\to }{p}\right)\right)={ℋ}^{s}\left(E\right),\phantom{\rule{0.277778em}{0ex}}{𝒫}^{s}\left(E\left(\stackrel{\to }{p}\right)\right)={𝒫}^{s}\left(E\right),$moreover both of ${ℋ}^{s}\left(E\right)$ and ${𝒫}^{s}\left(E\right)$ are finite positive;(ii) If $\stackrel{\to }{p}$ is a positive probability vector other than $\left({r}_{0}^{s},{r}_{1}^{s},\cdots ,{r}_{m-1}^{s}\right)$, then the gauge functions can be partitioned as follows${ℋ}^{g}\left(E\left(\stackrel{\to }{p}\right)\right)=+\infty ⇔\underset{t\to 0}{\overline{\mathrm{lim}}}\frac{logg\left(t\right)}{logt}\le \alpha ;\phantom{\rule{4pt}{0ex}}{ℋ}^{g}\left(E\left(\stackrel{\to }{p}\right)\right)=0⟺\underset{t\to 0}{\overline{\mathrm{lim}}}\frac{logg\left(t\right)}{logt}>\alpha ,$$...$

Page 1 Next