Metrical theorems on fractional parts of sequences

Abstract

i) 1. Introduction. Let C be the additive group of real numbers modulo 1, and let x -* {x} be the natural mapping from the reals onto C. It is clear what we shall mean by an interval / in C and by the length 1(1) of /. Denote the distance of the real number a to the closest integer by || a ||. The image in C of the set of reals satisfying || -01| ^ s with given 0 and 0 < < 1/2 is an example of an interval