Abstract
The sequence of integersn 1<n 2<n 3<... is said to be homogeneously distributed if\(\mathop {\lim }\limits_{m \to + \infty } (1/m)\sum\limits_{k = 1}^m {\exp (2\pi in_k \alpha )} = 0\) for all non-integral real α. The existence of such sequences with a prescribed subexponential growth is shown, the recurrent properties of these sequences are discussed.
Similar content being viewed by others
References
Erdös, P., Taylor, S. J.: On the set of points of convergence of a lacunary trigonometric series and the equidistribution properties of related sequences. Proc. London Math. Soc. (3),7, 598–615 (1957).
Furstenberg, H.: Poincaré recurrence and number theory. Bull. Amer. Math. Soc.5, 211–234 (1981).
Furstenberg, H.: Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton, N. J.: Univ. Press. 1981.
Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. New York: Wiley-Interscience. 1974.
Karacuba, A. A.: Estimates for trigonometric sums. Proc. Steklov Inst.112, 251–265 (1971).
Niederreiter, H.: On a paper of Blum, Eisenberg and Hahn concerning ergodic theory and the distribution of sequences in the Bohr group. Acta Sci. Math.37, 103–108 (1975).
Pollington, A. D.: On the density of sequence {n k ζ}. Ill. J. Math.23, 151–155 (1979).
Veech, W. A.: Well distributed sequences of integers. Trans. Amer. Math. Soc.161, 63–70 (1971).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Boshernitzan, M. Homogeneously distributed sequences and Poincaré sequences of integers of sublacunary growth. Monatshefte für Mathematik 96, 173–181 (1983). https://doi.org/10.1007/BF01605486
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01605486