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 $y_n\equiv\displaystyle\sum^{n-1}_{k=0}＼χi_I(\tau^kx)$ (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 $\displaystyle\sum^N_{j=1}c_j2^{-j}$, $c_j\in＼{0,1＼}$.
$\tau_2$를 $\tau_2x$ = 2x (mod 1)인 에르고딕 측도보존변환이라고 하자. KroneckerWeyl 모듈로 2 정리와 관계하여 $y_n \equiv \displaystyle\sum^{n-1}_{k=0} \chi_I(\tau^k_2x) (mod 2)$일때 a,b가 유한소수로서 간격 I=[a,b]가 [0,$\frac{3}{4}$]이나 [$\frac{1}{4}$, 1]의 부분집합이면 $\frac{1}{N} \displaystyle\sum^{N}_{n=1} y_n$의 극한값이 $\frac{1}{2}$ 임을 보였다.
