# On functions with bounded remainder

• Volume: 39, Issue: 1, page 17-26
• ISSN: 0373-0956

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## Abstract

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Let $T:ℝ/ℤ\to ℝ/ℤ$ be a von Neumann-Kakutani $q$- adic adding machine transformation and let $\varphi \in {C}^{1}\left(\left[0,1\right]\right)$. Put${\varphi }_{n}\left(x\right):=\varphi \left(x\right)+\varphi \left(Tx\right)+...+\varphi \left({T}^{n-1}x\right),\phantom{\rule{4pt}{0ex}}x\in ℝ/ℤ,\phantom{\rule{4pt}{0ex}}n\in ℕ.$We study three questions:1. When will $\left({\varphi }_{n}\left(x\right){\right)}_{n\ge 1}$ be bounded?2. What can be said about limit points of $\left({\varphi }_{n}\left(x\right){\right)}_{n\ge 1}?$3. When will the skew product $\left(x,y\right)↦\left(Tx,y+\varphi \left(x\right)\right)$ be ergodic on $ℝ/ℤ×ℝ?$

## How to cite

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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 -

## References

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1. [1] Y. DUPAIN and V.T. SÓS, On the one-sided boundedness of discrepancy-function of the sequence {nα}, Acta Arith., 37 (1980), 363-374. Zbl0445.10041MR82c:10058
2. [2] H. FAURE, Etude des restes pour les suites de Van der Corput généralisées, J. Number Th., 16 (1983), 376-394. Zbl0513.10047MR84g:10082
3. [3] W.H. GOTTSCHALK and G.A. HEDLUND, Topological Dynamics, AMS Colloq. Publ., 1955. Zbl0067.15204MR17,650e
4. [4] P. HELLEKALEK, Regularities in the distribution of special sequences, J. Number Th., 18 (1984), 41-55. Zbl0531.10055MR85e:11052
5. [5] P. HELLEKALEK, Ergodicity of a class of cylinder flows related to irregularities of distribution, Comp. Math., 61 (1987), 129-136. Zbl0619.10051MR88g:28018
6. [6] P. HELLEKALEK and G. LARCHER, On the ergodicity of a class of skew products, Israel J. Math., 54 (1986), 301-306. Zbl0609.28007MR87k:28013
7. [7] L.K. HUA and Y. WANG, Applications of number theory to numerical analysis, Springer-Verlag, Berlin, New York, 1981. Zbl0465.10045MR83g:10034
8. [8] H. KESTEN, On a conjecture of Erdös and Szüsz related to uniform distribution mod 1, Acta Arith., 12 (1966), 193-212. Zbl0144.28902MR35 #155
9. [9] L. KUIPERS and H. NIEDERREITER, Uniform distribution of sequences, John Wiley & Sons, New York, 1974. Zbl0281.10001MR54 #7415
10. [10] I. OREN, Ergodicity of cylinder flows arising from irregularities of distribution, Israel J. Math., 44 (1983), 127-138. Zbl0563.28010MR84i:10055
11. [11] K. PETERSEN, On a series of cosecants related to a problem in ergodic theory, Comp. Math., 26 (1973), 313-317. Zbl0269.10030MR48 #4273

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