Convergence Results for r-Iterated Means of the Denominators of the Lüroth Series

Giuliano, Rita

doi:10.1515/udt-2016-0020

Abstract

In the present paper we extend two classic asymptotic results concerning convergence in probability and convergence in distribution for the denominators of the Lüroth series and obtain new theorems concerning the same two kinds of convergence for the r-iterated arithmetic means of such denominators. These results are extended to r-iterated weighted means.

References

[1] BARREIRA, L.—IOMMI, G.: Frequency of digits in the Lüroth expansion, J.Number Theory 129 (2009), 1479–1490.10.1016/j.jnt.2008.06.002
Search in Google Scholar Back to article
[2] BARRIONUEVO, J.—BURTON, R. M.—DAJANI, K.—KRAAIKAMP, C.: Ergodic properties of generalized Lüroth series, Acta Arith. 74 (1996), no. 4, 311–327.
Search in Google Scholar Back to article
[3] CAO, C.—WU, J.—ZHANG, Z.: The efficiency of approximating real numbers by Lüroth expansion, Czechoslovak Math. J. 63 (138) (2013), no. 2, 497–513.
Search in Google Scholar Back to article
[4] GALAMBOS, J.: Representations of Real Numbers by Infinite Series, in: Lect. Notes in Math. Vol. 502, Springer-Verlag, Berlin, Heidelberg, New York, 1976, vi +146 pp.10.1007/BFb0081642
Search in Google Scholar Back to article
[5] JAGER, H.—DE VROEDT, C.: Lüroth series and their ergodic properties, Indag. Math. 31 (1968), 31–42.
Search in Google Scholar Back to article
[6] KALPAZIDOU, S.—KNOPFMACHER A.—KNOPFMACHER J.: Metric properties of alternating Lüroth series, Port. Math. 48 (1991), 319–325.
Search in Google Scholar Back to article
[7] KALPAZIDOU, S.—KNOPFMACHER, A.—KNOPFMACHER, J.: Lüroth-type alternating series representations for real numbers, Acta Arith. 55 (1990), no. 4, 311–322.
Search in Google Scholar Back to article
[8] KESSEBHMER, M.—MUNDAY, S.—STRATMANN, B. O.: Strong renewal theorems and Lyapunov spectra for α-Farey and α-Lüroth systems, Ergodic Theory Dynam. Systems 32 (2012), no. 3, 989–1017.
Search in Google Scholar Back to article
[9] LINERO, A.—ROSALSKY, A.: On the Toeplitz lemma, convergence in probability, and mean convergence, Stoch. Anal. Appl., 31 (2013), no. 4, 684–694.
Search in Google Scholar Back to article
[10] LÜROTH, J.: Ueber eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe, Math. Ann. 21 (1883), 411–423.10.1007/BF01443883
Search in Google Scholar Back to article
[11] MUNDAY, S.: (2011) A note on Diophantine fractals for α–Lüroth systems, Integers 11B, Paper No. A10, 14 pp.
Search in Google Scholar Back to article
[12] PERRON, O.: Irrationalzahlen, de Gruyter, Berlin, 1960.10.1515/9783110836042
Search in Google Scholar Back to article
[13] ŠALÁT, T.: Zur metrischen Theorie der Lürothschen Entwicklungen der reellen Zahlen, Czechoslovak Math. J. 18 (1968), 489–522.10.21136/CMJ.1968.100848
Search in Google Scholar Back to article
[14] SHEN, L.—WU, J.: (2007) On the error-sum function of Lüroth series, J. Math. Anal. Appl. 329 no. 2, 1440–1445.
Search in Google Scholar Back to article
[15] VERVAAT, W.: Success Epochs in Bernoulli Trials: with Applications to Number Theory, Math. Centre Tracts Vol. 42, Amsterdam, 1972. iii+166 pp, (loose errata).
Search in Google Scholar Back to article

Convergence Results for r-Iterated Means of the Denominators of the Lüroth Series

Abstract

Paradigm

My account