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Mod 2 normal numbers and skew products

Geon Ho Choe; Toshihiro Hamachi; Hitoshi Nakada — 2004

Studia Mathematica

Let E be an interval in the unit interval [0,1). For each x ∈ [0,1) define dₙ(x) ∈ 0,1 by $d ₙ (x) : = \sum_{i = 1}^{n} 1_{E} (2^{i - 1} x) (m o d 2)$ , 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 ≠(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.

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