Irregularities of distribution

Abstract: The term “Irregularities of distribution” appeared for the first time in the title of 1954 K. Roth’s seminal paper and referred to a conjecture of J. van der Corput on the non-existence of a “good” way to choose an infinite sequence in the unit interval. Roth approached van der Corput’s conjecture by checking the quality of any choice of N points in the 2-dimensional torus with respect to arbitrary squares therein, and proving a logarithmic lower bound for the discrepancy. Later W. Schmidt, H. Montgomery and J. Beck independently proved that the discrepancy is at least a power of N when squares are replaced with disks.

We construct a family of intermediate cases and we show that positive curvature plays no role in this problem which reduces to a careful study of the decay of certain Fourier transforms.

We shall also describe two related d-dimensional problems.

(from joint works with Luca Brandolini and Leonardo Colzani)

classical analysis and ODEsfunctional analysisrepresentation theoryspectral theory

Audience: researchers in the topic

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