Diophantine approximation generalized

In this paper we study the set of $x\in[0,1]$ for which the inequality $|x-x_n|<z_n$ holds for infinitely many $n=1,2,\dots$. Here $x_n\in[0,1)$ and $z_n>0$, $z_n\to0$, are sequences. In the first part of the paper we summarize known results. In the second part, using the theory of distribution functions of sequences, we find the asymptotic density of $n$ for which $|x-x_n|<z_n$, where $x$ is a discontinuity point of some distribution function of $x_n$. Generally, we also prove, for an arbitrary sequence $x_n$, that there exists $z_n$ such that the density of $n=1,2,\dots$, $x_n\to x$, is the same as the density of $n=1,2,\dots$, $|x-x_n|<z_n$, for $x\in[0,1]$. Finally we prove, using the longest gap $d_n$ in the finite sequence $x_1,x_2,\dots,x_n$, that if $d_n\le z_n$ for all $n$, $z_n\to0$, and $z_n$ is non-increasing, then $|x-x_n|<z_n$ holds for infinitely many $n$ and for almost all $x\in[0,1]$.