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Van der Corput sets in \(\mathbb Z^d\). (English) Zbl 1177.37018

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).

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
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