Geodesic
On recurrence property of Riesz--Raikov sums
Lobachevskii journal of mathematics, Tome 26 (2007), pp. 27-31.

Voir la notice de l'article provenant de la source Math-Net.Ru

The Riesz–Raikov sums $\sum f(\theta^kx)$ are recurrent in most cases.
Keywords: recurrence, lacunary series, Riesz–Raikov sums, non-conventional averages, mixing central limit theorem. 
@article{LJM_2007_26_a3,
     author = {K. Fukuyama and R. Kondo},
     title = {On recurrence property of {Riesz--Raikov} sums},
     journal = {Lobachevskii journal of mathematics},
     pages = {27--31},
     publisher = {mathdoc},
     volume = {26},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/LJM_2007_26_a3/}
}
TY  - JOUR
AU  - K. Fukuyama
AU  - R. Kondo
TI  - On recurrence property of Riesz--Raikov sums
JO  - Lobachevskii journal of mathematics
PY  - 2007
SP  - 27
EP  - 31
VL  - 26
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/LJM_2007_26_a3/
LA  - en
ID  - LJM_2007_26_a3
ER  - 
%0 Journal Article
%A K. Fukuyama
%A R. Kondo
%T On recurrence property of Riesz--Raikov sums
%J Lobachevskii journal of mathematics
%D 2007
%P 27-31
%V 26
%I mathdoc
%U http://geodesic.mathdoc.fr/item/LJM_2007_26_a3/
%G en
%F LJM_2007_26_a3
K. Fukuyama; R. Kondo. On recurrence property of Riesz--Raikov sums. Lobachevskii journal of mathematics, Tome 26 (2007), pp. 27-31. http://geodesic.mathdoc.fr/item/LJM_2007_26_a3/

[1] J. M. Anderson, D. Pitt, “On recurrence properties of certain lacunary series. I. general results”, Jour. reine angewandt. Math., 377 (1987), 65–82 ; “II. the series $\sum_{k=1}^n\exp(ia^k\theta)$”, 83–96 | MR | Zbl

[2] K. Fukuyama, “The central limit theorem for Riesz-Raikov sums”, Prob. Theory Related Fields, 100 (1994), 57–75 | DOI | MR | Zbl

[3] K. Fukuyama, “The central limit theorem for $\sum f(\theta^nx)g(\theta^{n^2}x)$”, Ergodic Theory Dynamical Systems, 20 (2000), 1335–1353 | DOI | MR | Zbl

[4] D. J. Grubb, C. N. Moore, “Certain lacunary cosine series are recurrent”, Studia Math., 108 (1994), 21–23 | MR | Zbl

[5] J. Hawkes, “Probabilistic behaviour of some lacunary series”, Z. Wahr. verw. Geb., 53 (1980), 21–33 | DOI | MR | Zbl

[6] I. A. Ibragimov, “The central limit theorem for sums of functions of independent variables and sums of the form $\sum f(2^kt)$”, Th. Prob. Appl., 12 (1967), 596–607 | DOI | MR | Zbl

[7] M. Kac, “On the distribution of values of the type $\sum f(2^kt)$”, Ann. Math., 47 (1946), 33–49 | DOI | MR | Zbl

[8] B. Petit, “Le théorème limite central pour des sommes de Riesz-Raikov”, Probab. Theory Relat. Fields, 93 (1992), 407–438 | DOI | MR | Zbl

[9] S. Takahashi, “The law of the iterated logarithm for a gap sequence with infinite gaps”, Tôhoku Math. J. (2), 15 (1963), 281–288 | DOI | MR | Zbl

[10] D. Ullrich, “Recurrence for lacunary cosine series”, Contemp. Math., 137 (1992), 459–467 | MR | Zbl