On normal numbers mod 2

Abstract

It is proved that a real-valued function f(x)=exp(πiχI(x)), where I is an interval contained in [0,1), is not of the form f(x)=q(2x)−−−−q(x) with |q(x)|=1 a.e. if I has dyadic endpoints. A relation of this result to the uniform distribution mod 2 is also shown.