Israel Journal of Mathematics

, Volume 59, Issue 2, pp 150–166

Ruzsa’s problem on sets of recurrence


  • J. Bourgain
    • Institut des Hautes Etudes Scientifiques

DOI: 10.1007/BF02787258

Cite this article as:
Bourgain, J. Israel J. Math. (1987) 59: 150. doi:10.1007/BF02787258


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={nZ;\(\hat \mu \)(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.

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© Hebrew University 1987