Bergelson, Vitaly; Lesigne, Emmanuel Van der Corput sets in \(\mathbb Z^d\). (English) Zbl 1177.37018 Colloq. Math. 110, No. 1, 1-49 (2008). This is a beautiful survey, a collection of generalizations and a collection of open problems in one. Below we give a small selection.A classical van der Corput set is a set \(D\) of integers with the property that the uniform distribution of a real sequence \((x_n)\) modulo one follows from the u.d. of the differences \((x_{n+d}-x_n)\) for all \(d\in D\). This property is connected with recurrence, difference sets, positive cosine polynomials and continuity of measures with a prescribed spectrum.All these concepts can be generalized to subsets of \(\mathbb Z^d\). Some of the proofs can also quite naturally be extended to the multidimensional case, some not so easily. The following property (Th. 1.32) is new even for the classical case: \(D\) is a van der Corput set if and only if any positive measure \(\sigma \) on the torus such that \(\sum _{d\in D} |\hat \sigma (d)| <\infty \) is continuous.An enhanced v.d.C. set is a set \(D\) of integers with the following property. If \((u_n)\) is a sequence of complex numbers with modulus 1 such that \[ \gamma (d) = \lim N^{-1} \sum _{n=1}^N u_{n+d} \overline u_n \] exists for \(d\in D\) and \(\to 0\) as \(d\to \infty \), then necessarily \( \lim N^{-1} \sum _{n=1}^N u_n =0 \). Usual v.d.C. sets use the stronger assumption that \(\gamma (d)=0\) for all \(d\in D\). This property is shown to be equivalent to the FC\({}^+\) property, that is, to that every measure satisfying \(\hat \sigma (d)\to 0\) is continuous. The main unsolved problem here is whether this concept is really stronger than the simple v.d.C.; the authors expect a positive answer. No candidate of example was found among the natural sets investigated (say, polynomials). Reviewer: Imre Z. Ruzsa (Budapest) Cited in 3 ReviewsCited in 13 Documents MSC: 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) 11B05 Density, gaps, topology 11K06 General theory of distribution modulo \(1\) 28D05 Measure-preserving transformations 42A61 Probabilistic methods for one variable harmonic analysis 42A82 Positive definite functions in one variable harmonic analysis Keywords:van der Corput set; set of recurrence; intersective set; uniform distribution; positive definite sequence; continuity of measure PDFBibTeX XMLCite \textit{V. Bergelson} and \textit{E. Lesigne}, Colloq. Math. 110, No. 1, 1--49 (2008; Zbl 1177.37018) Full Text: DOI arXiv