On functions with bounded remainder

Hellekalek, P., and Larcher, Gerhard. "On functions with bounded remainder." Annales de l'institut Fourier 39.1 (1989): 17-26. <http://eudml.org/doc/74823>.

@article{Hellekalek1989,
abstract = {Let $T: \{\Bbb R\}/\{\Bbb Z\}\rightarrow \{\Bbb R\}/\{\Bbb Z\}$ be a von Neumann-Kakutani $q$- adic adding machine transformation and let $\phi \in C^1([0,1])$. Put\begin\{\}\phi \_n(x):=\phi (x)+\phi (Tx)+\ldots \{\}+\phi (T^\{n-1\}x),\ x\in \{\Bbb R\}/\{\Bbb Z\},\ n\in \{\Bbb N\}.\end\{\}We study three questions:1. When will $(\phi _ n(x))_\{n\ge 1\}$ be bounded?2. What can be said about limit points of $(\phi _ n(x))_\{n\ge 1\}?$3. When will the skew product $(x,y)\mapsto (Tx,y+\phi (x))$ be ergodic on $\{\Bbb R\}/\{\Bbb Z\}\times \{\Bbb R\}?$},
author = {Hellekalek, P., Larcher, Gerhard},
journal = {Annales de l'institut Fourier},
keywords = {ergodicity; q-adic transformation; functions with bounded remainder; uniform distribution; discrepancy; von Neumann-Kakutani q-adic adding machine transformation},
language = {eng},
number = {1},
pages = {17-26},
publisher = {Association des Annales de l'Institut Fourier},
title = {On functions with bounded remainder},
url = {http://eudml.org/doc/74823},
volume = {39},
year = {1989},
}

TY - JOUR
AU - Hellekalek, P.
AU - Larcher, Gerhard
TI - On functions with bounded remainder
JO - Annales de l'institut Fourier
PY - 1989
PB - Association des Annales de l'Institut Fourier
VL - 39
IS - 1
SP - 17
EP - 26
AB - Let $T: {\Bbb R}/{\Bbb Z}\rightarrow {\Bbb R}/{\Bbb Z}$ be a von Neumann-Kakutani $q$- adic adding machine transformation and let $\phi \in C^1([0,1])$. Put\begin{}\phi _n(x):=\phi (x)+\phi (Tx)+\ldots {}+\phi (T^{n-1}x),\ x\in {\Bbb R}/{\Bbb Z},\ n\in {\Bbb N}.\end{}We study three questions:1. When will $(\phi _ n(x))_{n\ge 1}$ be bounded?2. What can be said about limit points of $(\phi _ n(x))_{n\ge 1}?$3. When will the skew product $(x,y)\mapsto (Tx,y+\phi (x))$ be ergodic on ${\Bbb R}/{\Bbb Z}\times {\Bbb R}?$
LA - eng
KW - ergodicity; q-adic transformation; functions with bounded remainder; uniform distribution; discrepancy; von Neumann-Kakutani q-adic adding machine transformation
UR - http://eudml.org/doc/74823
ER -