Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A metric discrepancy result for lacunary sequences


Authors: Katusi Fukuyama and Tetsujin Watada
Journal: Proc. Amer. Math. Soc. 140 (2012), 749-754
MSC (2010): Primary 11K38; Secondary 60F15
DOI: https://doi.org/10.1090/S0002-9939-2011-10940-7
Published electronically: June 23, 2011
MathSciNet review: 2869060
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that every value greater than or equal to $ 1/2 $ can be a constant appearing in the law of the iterated logarithm for discrepancies of a lacunary sequence satisfying the Hadamard gap condition.


References [Enhancements On Off] (What's this?)

  • 1. C. Aistleitner, On the law of the iterated logarithm for the discrepancy of lacunary sequences, Trans. Amer. Math. Soc., 362 (2010) 5967-5982. MR 2661504
  • 2. C. Aistleitner, I. Berkes, R. Tichy, On permutation of Hardy-Littlewood-Polya sequences, Trans. Amer. Math. Soc. (to appear).
  • 3. I. Berkes, W. Philipp, The size of trigonometric and Walsh series and uniform distribution mod 1, Jour. London Math. Soc. (2), 50 (1994) 454-464. MR 1299450 (96e:11099)
  • 4. K. Fukuyama, An asymptotic property of gap series, Acta Math. Hungar., 97 (2002) 257-264. MR 1933732 (2003i:42013)
  • 5. K. Fukuyama, The law of the iterated logarithm for discrepancies of $ \{\theta^n x\}$, Acta Math. Hungar., 118 (2008) 155-170. MR 2378547 (2008m:60049)
  • 6. K. Fukuyama, K. Nakata, A metric discrepancy result for the Hardy-Littlewood-Pólya sequences, Monatsh. Math., 160 (2010) 41-49. MR 2610311
  • 7. K. Fukuyama, A central limit theorem and a metric discrepancy result for sequence with bounded gaps, Dependence in Probability, Analysis and Number Theory, A volume in memory of Walter Philipp, Kendrick Press, Heber City, UT (2010), 233-246.
  • 8. K. Fukuyama, N. Hiroshima, Metric discrepancy results for subsequences of $ \{\theta^k x\}$, Monatsh. Math. (to appear).
  • 9. W. Philipp, Limit theorems for lacunary series and uniform distribution mod 1, Acta Arith., 26 (1975) 241-251. MR 0379420 (52:325)
  • 10. W. Philipp, A functional law of the iterated logarithm for empirical distribution functions of weakly dependent random variables, Ann. Probab., 5 (1977) 319-350. MR 0443024 (56:1397)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11K38, 60F15

Retrieve articles in all journals with MSC (2010): 11K38, 60F15


Additional Information

Katusi Fukuyama
Affiliation: Department of Mathematics, Kobe University, Rokko, Kobe, 657-8501 Japan
Email: fukuyama@math.kobe-u.ac.jp

Tetsujin Watada
Affiliation: Department of Mathematics, Kobe University, Rokko, Kobe, 657-8501 Japan

DOI: https://doi.org/10.1090/S0002-9939-2011-10940-7
Keywords: Discrepancy, lacunary sequence, law of the iterated logarithm
Received by editor(s): November 26, 2010
Received by editor(s) in revised form: December 8, 2010
Published electronically: June 23, 2011
Additional Notes: The first author was supported in part by KAKENHI 19204008.
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society