Distribution functions of ratio sequences. An expository paper

Abstract

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.