Abstract
The main purpose of this paper is to prove the existence of Poincaré sequences of integers which are not van der Corput sets. This problem was considered in I. Ruzsa’s expository article [R1] (1982–83) on correlative and intersective sets. Thus the existence is shown of a positive non-continuous measureμ on the circle which Fourier transform vanishes on a set of recurrence, i.e.S={n ∈Z;\(\hat \mu \)(n)=0} is a set of recurrence but not a van der Corput set. The method is constructive and involves some combinatorial considerations. In fact, we prove that the generic density condition for both properties are the same.
Similar content being viewed by others
References
A. Bertrand-Mathis,Ensembles intersectifs et récurrence de Poincaré, Isr. J. Math.55 (1986), 184–198.
H. Furstenberg,Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, 1981.
T. Kamae and M. Mendès France,Van der Corput’s difference theorem, Isr. J. Math.31 (1978), 335–342.
Y. Katznelson,Sequences of integers dense in the Bohr group, Proc. Royal Inst. Techn. Stockholm 76–86 (1973).
Y. Katznelson,Suites aléatoires d’entiers, Lecture Notes in Math.336, Springer-Verlag, Berlin, 1973, pp. 148–152.
J. Lopez and K. Ross,Sidon sets, Lecture notes in Pure and Applied Math., No. 13, M. Dekker, New York, 1975.
Y. Peres, Master Thesis.
I. Ruzsa,Ensembles intersectifs, Séminaire de Théorie des Nombres de Bordeaux, 1982–83.
I. Ruzsa,On difference sets, Stud. Sci. Math. Hung.13 (1978), 319–326.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bourgain, J. Ruzsa’s problem on sets of recurrence. Israel J. Math. 59, 150–166 (1987). https://doi.org/10.1007/BF02787258
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02787258