Distribution functions of ratio sequences. An expository paper

Rok, strany: 2015, 133 - 185

This expository paper presents known results on distribution functions $g(x)$ of the sequence of blocks $X_n=\bl(((x₁) / (x_n)),((x₂) / (x_n)),…,((x_n) / (x_n))\br)$, $n=1,2,…$, where $x_n$ is an increasing sequence of positive integers. Also presents results of the set $G(X_n)$ of all distribution functions $g(x)$. Specially: \begin{itemize} \item[–] continuity of $g(x)$; \item[–] connectivity of $G(X_n)$; \item[–] singleton of $G(X_n)$; \item[–] one-step $g(x)$; \item[–] uniform distribution of $X_n$, $n=1,2,…$; \item[–] lower and upper bounds of $g(x)$; \item[–] applications to bounds of $((1) / (n))∑_i=1ⁿ((x_i) / (x_n))$; \item[–] many examples, e.g., $ X_n=\bl(((2) / (p_n)),((3) / (p_n)),…,((p_n-1) / (p_n)), ((p_n) / (p_n))\br)$, where $p_n$ is the $n$th prime, is uniformly distributed. \end{itemize} The present results have been published by 25 papers of several authors between 2001–2013.

ISO 690:

Strauch, O. 2015. Distribution functions of ratio sequences. An expository paper. In Tatra Mountains Mathematical Publications, vol. 64, no.3, pp. 133-185. 1210-3195.

APA:

Strauch, O. (2015). Distribution functions of ratio sequences. An expository paper. Tatra Mountains Mathematical Publications, 64(3), 133-185. 1210-3195.