Mod 2 normal numbers and skew products

Geon Ho Choe, Toshihiro Hamachi, and Hitoshi Nakada. "Mod 2 normal numbers and skew products." Studia Mathematica 165.1 (2004): 53-60. <http://eudml.org/doc/286569>.

@article{GeonHoChoe2004,
abstract = {Let E be an interval in the unit interval [0,1). For each x ∈ [0,1) define dₙ(x) ∈ 0,1 by $dₙ(x) := ∑_\{i=1\}^\{n\} 1_\{E\} (\{2^\{i-1\}x\}) (mod 2)$, 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.},
author = {Geon Ho Choe, Toshihiro Hamachi, Hitoshi Nakada},
journal = {Studia Mathematica},
keywords = {ergodicity; mod 2 normal number; skew product; coboundary},
language = {eng},
number = {1},
pages = {53-60},
title = {Mod 2 normal numbers and skew products},
url = {http://eudml.org/doc/286569},
volume = {165},
year = {2004},
}

TY - JOUR
AU - Geon Ho Choe
AU - Toshihiro Hamachi
AU - Hitoshi Nakada
TI - Mod 2 normal numbers and skew products
JO - Studia Mathematica
PY - 2004
VL - 165
IS - 1
SP - 53
EP - 60
AB - Let E be an interval in the unit interval [0,1). For each x ∈ [0,1) define dₙ(x) ∈ 0,1 by $dₙ(x) := ∑_{i=1}^{n} 1_{E} ({2^{i-1}x}) (mod 2)$, 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.
LA - eng
KW - ergodicity; mod 2 normal number; skew product; coboundary
UR - http://eudml.org/doc/286569
ER -