### 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 |
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Pages (from-to) | 53-60 |

Number of pages | 8 |

Journal | Studia Mathematica |

Volume | 165 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2004 Jan 1 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Choe, G. H., Hamachi, T., & Nakada, H. (2004). Mod 2 normal numbers and skew products.

*Studia Mathematica*,*165*(1), 53-60. https://doi.org/10.4064/sm165-1-4