One-dimensional quasi-uniform Kronecker sequences

Abstract

AbstractIn this short note, we prove that the one-dimensional Kronecker sequence $$i\alpha \bmod 1, i=0,1,2,\ldots ,$$ i α mod 1 , i = 0 , 1 , 2 , … , is quasi-uniform over the unit interval [0, 1] if and only if $$\alpha $$ α is a badly approximable number. Our elementary proof relies on a result on the three-gap theorem for Kronecker sequences due to Halton (Proc Camb Philos Soc, 61:665–670, 1965).