Mod 2 normal numbers and skew products

Abstract

Let E be an interval in the unit interval [0, 1). For each x is an element of [0, 1) define d(n)(x) is an element of {0, 1} by d(n)(x) := Sigma(i=1)(n) 1(E)({2(i-1)x}) (mod 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 Sigma(n=1)(N) d(n)(x) converges to 1/2. It is shown that for any interval E not equal (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 Sigma(n=1)(N) d(n) (x) converges a.e. and the limit equals 1/3 or 2/3 depending on x.