# Mod 2 normal numbers and skew products

Geon Ho Choe, Toshihiro Hamachi, Hitoshi Nakada

Research output: Contribution to journalArticle

1 Citation (Scopus)

### Abstract

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=1n 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-1n=1N 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-1n=1N dn (x) converges a.e. and the limit equals 1/3 or 2/3 depending on x.

Original language English 53-60 8 Studia Mathematica 165 1 https://doi.org/10.4064/sm165-1-4 Published - 2004 Jan 1

### Keywords

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

### ASJC Scopus subject areas

• Mathematics(all)

• ## 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