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Mod 2 normal numbers and skew products

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Choe G., Hamachi T., Nakada H.

Studia Mathematica

2004

tom 165

nr 1

53-60

Let E be an interval in the unit interval [0,1). For each x ∈ [0,1) define dₙ(x) ∈ {0,1} by $dₙ(x) := ∑_{i=1}^{n} 1_{E} ({2^{i-1}x}) (mod 2)$, where t is the fractional part of t. Then x is called a normal number mod 2 with respect to E if $N^{-1} ∑_{n=1}^{N} dₙ(x)$ 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 $N^{-1} ∑_{n=1}^{N} dₙ(x)$ converges a.e. and the limit equals 1/3 or 2/3 depending on x.

Ograniczanie wyników

1 Studia Mathematica

1 Choe G.

1 Hamachi T.

1 Nakada H.

1 2004

Wyniki wyszukiwania

Mod 2 normal numbers and skew products