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Zbl 1183.11042
Grekos, Georges; Strauch, Oto
Distribution functions of ratio sequences. II. (English)
[J] Unif. Distrib. Theory 2, No. 1, 53-77 (2007). ISSN 1336-913X

Let $x_n$, $n=1,2,\dots $ be an increasing sequence of positive integers. The second author and {\it J. T. Tóth} [Publ. Math. 58, 751--778 (2001; Zbl 0980.11031)] introduced and studied the sequence composed with blocks $X_n=\left (\frac {x_1}{x_n},\frac {x_2}{x_n},\dots ,\frac {x_n}{x_n}\right )$. For every block $X_n$ they define the step distribution function $F(X_n,x)=\frac {1}{n}\# \{i\le n:x_i/x_n<x\}$, and denote by $G(X_n)$ the set of all distribution functions of the sequence $X_n$, $n=1,2,\dots $ (i.e., the set of all possible weakly limits $\lim _{k\to \infty }F(X_{n_k},x)=g(x)$). The paper under review contains the following three main results: (i) The set $G(X_n)$ is not connected in general, contrary to the well known fact that the set $G(y_n)$ of all distribution functions of the sequence $y_n\in [0,1)$, $n=1,2,\dots $, is connected for an arbitrary $y_n$ [see {\it R. Winkler}, Math. Nachr. 186, 303--312 (1997; Zbl; 0876.11040)]. (ii) Let $c_\alpha (x)$ be the one-step distribution function with the step $1$ in $\alpha \in [0,1]$. Then $c_1(x)\in G(X_n)$ need not imply $c_0(x)\in G(X_n)$. (iii) Assume that $G(X_n)\subset \{c_\alpha (x):\alpha \in [0,1]\}$, then $c_0(x)\in G(X_n)$ and if $G(X_n)$ contains two different distribution functions, then also $c_1(x)\in G(X_n)$. The paper concludes with open problems, one of which is the question whether $G(X_n)\subset \{c_\alpha (x):\alpha \in [0,1]\}$ implies $G(X_n)=\{c_0(x)\}$.
[Ladislav Miš\'ik (Ostrava)]

MSC 2000:: *11K31 Special sequences
11K06 General theory of distribution modulo 1

Keywords: Distribution function; connectivity; ratio sequence; block sequence

Citations: Zbl 0980.11031

Cited in: Zbl 1203.11012 Zbl 1153.11040

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