### Abstract

Let E be an interval in the unit interval [0, 1). For each x ∈ [0, 1) define d_{n} (x) ∈ {0, 1} by d_{n}(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} 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} ∑_{n=1}
^{N} d_{n} (x) converges a.e. and the limit equals 1/3 or 2/3 depending on x.

Original language | English |
---|---|

Pages (from-to) | 53-60 |

Number of pages | 8 |

Journal | Studia Mathematica |

Volume | 165 |

Issue number | 1 |

Publication status | Published - 2004 |

Externally published | Yes |

### Fingerprint

### Keywords

- Coboundary
- Ergodicity
- Mod 2 normal number
- Skew product

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Studia Mathematica*,

*165*(1), 53-60.

**Mod 2 normal numbers and skew products.** / Choe, Geon Ho; Hamachi, Toshihiro; Nakada, Hitoshi.

Research output: Contribution to journal › Article

*Studia Mathematica*, vol. 165, no. 1, pp. 53-60.

}

TY - JOUR

T1 - Mod 2 normal numbers and skew products

AU - Choe, Geon Ho

AU - Hamachi, Toshihiro

AU - Nakada, Hitoshi

PY - 2004

Y1 - 2004

N2 - 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 1E({2i-1x}) (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.

AB - 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 1E({2i-1x}) (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.

KW - Coboundary

KW - Ergodicity

KW - Mod 2 normal number

KW - Skew product

UR - http://www.scopus.com/inward/record.url?scp=5644229918&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=5644229918&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:5644229918

VL - 165

SP - 53

EP - 60

JO - Studia Mathematica

JF - Studia Mathematica

SN - 0039-3223

IS - 1

ER -