This expository paper presents known results on distribution functions $g(x)$ of the sequence of blocks $X_n=\left(\frac{x_1}{x_n},\frac{x_2}{x_n},\dots,\frac{x_n}{x_n}\right),$ $n=1,2,\dots$, 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: \par continuity of $g(x)$; \par connectivity of $G(X_n)$; \par singleton of $G(X_n)$; \par one-step $g(x)$; \par uniform distribution of $X_n$, $n=1,2,\dots$; \par lower and upper bounds of $g(x)$; \par applications to bounds of $\frac{1}{n}\sum_{i=1}^n\frac{x_i}{x_n}$; \par many examples, e.g. $ X_n=\left(\frac{2}{p_n},\frac{3}{p_n},\dots,\frac{p_{n-1}}{p_n}, \frac{p_n}{p_n}\right), $ where $p_n$ is the $n$th prime, is uniformly distributed. \par\noindent The present results have been published by 25 papers of many authors between 2001--2013.
@article{413,
title = {Distribution functions of ratio sequences. An expository paper},
journal = {Tatra Mountains Mathematical Publications},
volume = {65},
year = {2016},
doi = {10.2478/tatra.v64i0.413},
language = {EN},
url = {http://dml.mathdoc.fr/item/413}
}
Strauch, Oto. Distribution functions of ratio sequences. An expository paper. Tatra Mountains Mathematical Publications, Tome 65 (2016) . doi : 10.2478/tatra.v64i0.413. http://gdmltest.u-ga.fr/item/413/