Mod 2 normal numbers and skew products

• Autorzy: Geon Ho Choe , Toshihiro Hamachi , Hitoshi Nakada
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
• 37A30: Ergodic theorems, spectral theory, Markov operators
• 28D05: Measure-preserving transformations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2004
• Tom: 165
• Numer: 1
• Strony: 53-60

Daty

wydano

2004

Twórcy

autor

Geon Ho Choe

• Department of Mathematics, Korea Advanced Institute, of Science and Technology, Daejeon, South Korea

autor

Toshihiro Hamachi

• Department of Mathematics, Kyushu University, Fukuoka, Japan

autor

Hitoshi Nakada

• Department of Mathematics, Keio University, Yokohama, Japan

Identyfikatory

DOI

10.4064/sm165-1-4