On functions with bounded remainder
Annales de l'Institut Fourier, Volume 39 (1989) no. 1, p. 17-26

Let T:// be a von Neumann-Kakutani q- adic adding machine transformation and let φC 1 ([0,1]). Put

φn(x):=φ(x)+φ(Tx)+...+φ(Tn-1x),x/,n.

We study three questions:

1. When will (φ n (x)) n1 be bounded?

2. What can be said about limit points of (φ n (x)) n1 ?

3. When will the skew product (x,y)(Tx,y+φ(x)) be ergodic on /×?

Soit T:// une transformation du type Neumann-Kakutani en base q et soit φC 1 ([0,1]). Posons, pour x/, n,

φn(x):=φ(x)+φ(Tx)++φ(Tn-1x).

Nous étudions les trois questions suivantes :

1. Pour la suite (φ n (x)) n1 : à quelles conditions sera-t-elle bornée ?

2. Que peut-on dire sur les points d’adhérence de (φ n (x)) n1 ?

3. Pour le produit croisé (x,y)(Tx,y+φ(x)) sur le cylindre /× : à quelles conditions sera-t-il ergodique ?

@article{AIF_1989__39_1_17_0,
     author = {Hellekalek, P. and Larcher, Gerhard},
     title = {On functions with bounded remainder},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {39},
     number = {1},
     year = {1989},
     pages = {17-26},
     doi = {10.5802/aif.1156},
     zbl = {0674.28007},
     mrnumber = {90i:28024},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1989__39_1_17_0}
}
Hellekalek, P.; Larcher, Gerhard. On functions with bounded remainder. Annales de l'Institut Fourier, Volume 39 (1989) no. 1, pp. 17-26. doi : 10.5802/aif.1156. http://www.numdam.org/item/AIF_1989__39_1_17_0/

[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. | MR 82c:10058 | Zbl 0445.10041

[2] H. Faure, Etude des restes pour les suites de Van der Corput généralisées, J. Number Th., 16 (1983), 376-394. | MR 84g:10082 | Zbl 0513.10047

[3] W.H. Gottschalk and G.A. Hedlund, Topological Dynamics, AMS Colloq. Publ., 1955. | MR 17,650e | Zbl 0067.15204

[4] P. Hellekalek, Regularities in the distribution of special sequences, J. Number Th., 18 (1984), 41-55. | MR 85e:11052 | Zbl 0531.10055

[5] P. Hellekalek, Ergodicity of a class of cylinder flows related to irregularities of distribution, Comp. Math., 61 (1987), 129-136. | Numdam | MR 88g:28018 | Zbl 0619.10051

[6] P. Hellekalek and G. Larcher, On the ergodicity of a class of skew products, Israel J. Math., 54 (1986), 301-306. | MR 87k:28013 | Zbl 0609.28007

[7] L.K. Hua and Y. Wang, Applications of number theory to numerical analysis, Springer-Verlag, Berlin, New York, 1981. | MR 83g:10034 | Zbl 0465.10045

[8] H. Kesten, On a conjecture of Erdös and Szüsz related to uniform distribution mod 1, Acta Arith., 12 (1966), 193-212. | MR 35 #155 | Zbl 0144.28902

[9] L. Kuipers and H. Niederreiter, Uniform distribution of sequences, John Wiley & Sons, New York, 1974. | MR 54 #7415 | Zbl 0281.10001

[10] I. Oren, Ergodicity of cylinder flows arising from irregularities of distribution, Israel J. Math., 44 (1983), 127-138. | MR 84i:10055 | Zbl 0563.28010

[11] K. Petersen, On a series of cosecants related to a problem in ergodic theory, Comp. Math., 26 (1973), 313-317. | Numdam | MR 48 #4273 | Zbl 0269.10030