Uniform distribution modulo 2 on the gauss transformation

Abstract

Let T be an ergodic transformation. One may ask whether $\lim_{N\to\infty}\frac{1}{N}\sum\limits^N_{n=1}y_n(x),\;y_n{x}=0\,\,\mbox{or}\,\, 1$, exists and, if so, what is the value where $y_n(x)=\sum^{n-1}_{k=0}\chi_B(T^kx)$(mod2) and B is a measurable set. The limit might not be equal to 1/2 if $\exp(\pi{i}\chi_B(x))$ is a coboundary. We show that, when B is a finite union of intervals of [0,1) with rational end points, $\exp(\pi{i}\chi_B)$ is not a coboundary about the Gauss transformation on [0,1).