Mod 2 normal numbers and skew products
Volume 165 / 2004
Studia Mathematica 165 (2004), 53-60
MSC: Primary 28D05, 37A30, 11K16.
DOI: 10.4064/sm165-1-4
Abstract
Let $E$ be an interval in the unit interval $[0,1)$. For each $x \in [0,1)$ define $d_n(x) \in \{0,1 \}$ by $d_n(x) := \sum_{i=1}^n 1_E (\{2^{i-1} x\}) \pmod 2$, where $\{t\}$ is the fractional part of $t$. Then $x$ is called a normal number mod $2$ with respect to $E$ if $N^{-1} \sum_{n=1}^N d_n(x)$ converges to $1/2$. It is shown that for any interval $E \not=(1/6, 5/6)$ a.e. $x$ is a normal number mod $2$ with respect to $E$. For $E = (1/6, 5/6)$ it is proved that $N^{-1} \sum_{n=1}^N d_n(x)$ converges a.e. and the limit equals $1/3$ or $2/3$ depending on $x$.
