Weyl’s theorems on uniform distribution and Kronecker’s theorem

Abstract

§ 1. Introduction. We have seen in Chapter III that to any given irrational number ξ, there correspond infinitely many rational numbers p/q, such that |ξ − p/q| <1/q². From this follows Dirichlet’s theorem that corresponding to any given irrational number ξ, there exist infinitely many pairs of integers p and q, such that qξ differs from p by as little as we please. For given ε, 0<ε<1, we consider the integer 1 + [1/ε]. Since there exist infinitely many rationals p/q, such that |qξ − pl<1/q, it follows that there exist infinitely many fractions p/q, with denominator q ≥ 1 + [1/ε], for which we have |qξ − p|<1/q<ε.