Diophantine approximation generalized

In this paper we study the set of x ∈ [0, 1] for which the inequality |x − xn| < zn holds for infinitely many n = 1, 2, .... Here xn ∈ [0, 1) and zn s> 0, zn → 0, 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 − xn| < zn, where x is a discontinuity point of some distribution function of xn. Generally, we also prove, for an arbitrary sequence xn, that there exists zn such that the density of n = 1, 2, ..., xn → x, is the same as the density of n = 1, 2, ..., |x − xn| < zn, for x ∈ [0, 1]. Finally we prove, using the longest gap dn in the finite sequence x1, x2, ..., xn, that if dn ≤ zn for all n, zn → 0, and zn is non-increasing, then |x − xn| < zn holds for infinitely many n and for almost all x ∈ [0, 1].