Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T17:07:12.407Z Has data issue: false hasContentIssue false
  • English
  • Français

Notes on Erdös-Turán inequality

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  09 April 2009

Yukio Ohkubo
Yukio Ohkubo
Affiliation:
Faculty of Economics Kagoshima Keizai University Shimofukumoto-cho Kagoshima-shi 891-0191 Japan e-mail: ohkubo@eco.kkis.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A new version of Erdös-Turán's inequality is described. The purpose of the present paper is to show that the inequality provides better upper bounds for the discrepancies of some sequences than usual Erdös-Turán's inequality.

Keywords

DiscrepancyErdös-Turán's inequalityexponential sumvan der Corput's method.

MSC classification

Secondary: 11K38: Irregularities of distribution, discrepancy
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 67 , Issue 1 , August 1999 , pp. 51 - 57
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Baker, R. C. and Harman, G., ‘Sequences with bounded logarithmic discrepancy’, Math. Proc. Cambridge Philos. Soc. 107 (1990), 213–225.CrossRefGoogle Scholar
[2]Drmota, M. and Tichy, R. F., Sequences, discrepancies and applications, Lecture Notes in Math. 1651 (Springer, Berlin, 1997).CrossRefGoogle Scholar
[3]Kuipers, L. and Niederreiter, H., Uniform distribution of sequences (Wiley, New York, 1974).Google Scholar
[4]Tichy, R. F. and Turnwald, G., ‘Logarithmic uniform distribution of (αν + β log ν)’, Tsukuba J. Math. 10 (1986), 351–366.Google Scholar
[5]Titchmarsh, E. C., The theory of the Riemann zeta-function, 2nd ed. revised by Heath-Brown, D. R. (Clarendon Press, Oxford, 1986).Google Scholar