Let E be an interval in the unit interval [0,1). For each x ∈ [0,1) define dₙ(x) ∈ 0,1 by , where t is the fractional part of t. Then x is called a normal number mod 2 with respect to E if 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 converges a.e. and the limit equals 1/3 or 2/3 depending on x.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm165-1-4,
author = {Geon Ho Choe and Toshihiro Hamachi and Hitoshi Nakada},
title = {Mod 2 normal numbers and skew products},
journal = {Studia Mathematica},
volume = {162},
year = {2004},
pages = {53-60},
zbl = {1080.11056},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm165-1-4}
}
Geon Ho Choe; Toshihiro Hamachi; Hitoshi Nakada. Mod 2 normal numbers and skew products. Studia Mathematica, Tome 162 (2004) pp. 53-60. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm165-1-4/