Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-06-30T23:53:25.776Z Has data issue: false hasContentIssue false
  • English
  • Français

The Distribution of Sequences Modulo 1

Published online by Cambridge University Press:  20 November 2018

Alan Zame
Alan Zame*
Affiliation:
University of Miami, Coral Gables, Florida
Rights & Permissions [Opens in a new window]

Extract

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.

In a recent paper (2), Helson and Kahane consider the problem of the existence of real numbers x such that the sequence (λn x) (when reduced modulo 1) is not summable by a given regular Toeplitz method, where (λn) is a lacunary sequence of positive real numbers. Thus, as an example, they show the existence of uncountably many x such that the sequence (θn x) does not have a distribution function modulo 1, where θ is some fixed number > 1.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 697 - 709
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Cigler, J. and Helmburg, G., Neuere Entwicklungen der Theorie der Gleichverteilung, Jber. Deutsch. Math. Verein., 64 (1962), 141–147.Google Scholar
2. Helson, H. and Kahane, J.-P., A Fourier method in diophantine problems, J. Analyse Math., 15 (1965), 245–262.Google Scholar
3. Koksma, J. F., Ein mengentheoretischer Satz über die Gleichverteilung mod. Eins, Composotio Math., 2 (1935), 250–258.Google Scholar
4. Pisot, C., Sur la répartition mod un, Acta Arith., vol. 2 (1939), 174–179.Google Scholar
5. Pyateckii, I., On the distribution functions of the fractional parts of an exponential function, Izv. Akad. Nauk, 15 (1951), 47–52 (in Russian).Google Scholar
6. Vijayaraghavan, T., On the fractional parts of the powers of a number, IV, J. Indian Math. Soc., 12 (1948), 33–39.Google Scholar
7. Weyl, H., Über die Gleichverteilung von Zahlen mod. Eins., Math. Ann., 77 (1916), 313–352.Google Scholar