Uniform distribution modulo 2

Abstract

Let $\tau$ be any ergodic measure preserving transformation and let $\tau_2$ be one sided shift on [0,1). In comparison with the Kronecker-Weyl theorem modulo 2, we investigate the existence and the value of the limit of $\frac{1}{N} \displaystyle\sum^N_1 y_n$, where $y_n\equiv\displaystyle\sum^{n-1}_{k=0} χ_I(\tau^kx)$ (mod 2), $y_n \in {0,1}$. The limit is equal $\frac{1}{2}$ if $exp(\pi i\chi_I(x))$ is not a coboundary. When $\tau$ = $\tau_2$, the limit is equal to $\frac{1}{2}$, if I = [a,b] is contained in $[0,\frac{3}{4}]$ or [$\frac{1}{4}$,1], where a, b are of the form $\displaystyle\sum^N_{j=1}c_j2^{-j}$, $c_j\in{0,1}$.