On statistical limit points

Authors:
P. Kostyrko, M. Macaj, T. Salát and O. Strauch

Journal:
Proc. Amer. Math. Soc. **129** (2001), 2647-2654

MSC (2000):
Primary 40A05, 11K31, 11B05

DOI:
https://doi.org/10.1090/S0002-9939-00-05891-3

Published electronically:
December 7, 2000

MathSciNet review:
1838788

Full-text PDF

Abstract | References | Similar Articles | Additional Information

The set of all statistical limit points of a given sequence is characterized as an -set. It is also characterized in terms of discontinuity points of distribution functions of .

**[BBK95]**D. BEREND, M. D. BOSHERNITZAN, AND G. KOLESNIK,*Distribution modulo of some oscillating sequences, II.*Israel J. Math.**92**(1995), 125-147. MR**96j:11105****[Fr93]**J. A. FRIDY,*Statistical limit points.*Proc. Amer. Math. Soc.**118**(1993), 1187-1192. MR**94e:40008****[HR66]**H. HALBERSTAM AND K. F. ROTH,*Sequences, I.*Clarendon Press, Oxford, 1966. MR**35:1565****[KN74]**L. KUIPERS AND H. NIEDERREITER,*Uniform Distribution of Sequences.*John Wiley & Sons, New York, 1974. MR**54:7415****[My93]**G. MYERSON,*A sampler of recent developments in the distribution of sequences.*Lect. Notes Pure Appl. Math.**147**(1993), 163-190. MR**94a:11112****[Pa84]**D. P. PARENT,*Exercises in Number Theory.*Problem Books in Mathematics, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1984. MR**86f:11002****[Sch59]**I. J. SCHOENBERG,*The integrability of certain functions and related summability methods.*Amer. Math. Monthly**66**(1959), 361-375. MR**21:3696****[St95]**O. STRAUCH,*Uniformly maldistributed sequences in a strict sense.*Monatsh. Math.**120**(1995), 153-164. MR**96g:11095****[St97]**O. STRAUCH,*On set of distribution functions of a sequence.*In: Proc. Conf. Analytic and Elementary Number Theory, In Honor of E. Hlawka's 80th Birthday, Vienna, July 18-20, 1996, Universität Wien and Universität für Bodenkultur, (W. G. Nowak and J. Schoißengeier, eds.) Vienna, 1997, 214-229.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
40A05,
11K31,
11B05

Retrieve articles in all journals with MSC (2000): 40A05, 11K31, 11B05

Additional Information

**P. Kostyrko**

Affiliation:
Department of Mathematics, Comenius University, SK-842 15 Bratislava, Slovakia

Email:
kostyrko@fmph.uniba.sk

**M. Macaj**

Affiliation:
Department of Mathematics, Comenius University, SK-842 15 Bratislava, Slovakia

Email:
macaj@fmph.uniba.sk

**T. Salát**

Affiliation:
Department of Mathematics, Comenius University, SK-842 15 Bratislava, Slovakia

**O. Strauch**

Affiliation:
Mathematical Institute of the Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava, Slovakia

Email:
strauch@mat.savba.sk

DOI:
https://doi.org/10.1090/S0002-9939-00-05891-3

Keywords:
Statistically convergent sequence,
statistical limit point,
asymptotic density,
distribution function

Received by editor(s):
December 12, 1998

Received by editor(s) in revised form:
January 10, 2000

Published electronically:
December 7, 2000

Additional Notes:
This work was supported in part by grants from VEGA No. 2/5123/98 and No. 1/4323/97

Communicated by:
David Preiss

Article copyright:
© Copyright 2000
American Mathematical Society