Discrepancy theory and harmonic analysis

Abstract

In the present survey we discuss various applications of methods and ideas of harmonic analysis in problems of geometric discrepancy theory and irregularities of distribution. A great number of analytic tools (exponential sums, Fourier series, Fourier transform, orthogonal expansions and wavelets, Riesz products, Littlewood-Paley theory, Carleson's theorem) have found applications in this area. Some of the methods have been used since the birth of the subject, while the more modern ideas are still paving their way into the field. We illustrate their applications by considering several standard topics in uniform distribution theory: Weyl's criterion, metric estimates for the discrepancy of sequences, discrepancy with respect to balls and rotated cubes, lower bounds for the discrepancy function. This exposition of Fourier-analytic techniques in discrepancy theory is intended for a broad mathematical readership.

Metric number theory, lacunary series and systems of dilated functions

“…Thus the a.s. convergence behavior of series of independent random variables and of lacunary trigonometric series are in perfect accordance. Many similar results of the same type exist: for example, by a classical result of Salem and Zygmund [40], under the gap condition (10) we have…”

“…instead of (cos 2πn k x) k≥1 . A striking result for this general setting is a theorem of Philipp [38], who confirmed the so-called Erdős-Gál conjecture by proving that under (10) we have…”

“…As a consequence of (30) (and using a trick to modify the argument around equation (26) in such a way to get a generalized GCD sum of the form (27) instead of (23)) one can show that for any f of bounded variation satisfying (14) and for any strictly increasing sequence (n k ) k≥1 of positive integers the Carleson-type inequality Concluding remark: There exists a close connection between discrepancy theory and harmonic analysis, which we have for example observed in Weyl's criterion (2), in the Erdős-Turán inequality (8) and in the Carleson convergence theorem in Section 5. This connection goes far beyond the material contained in this article, and is comprehensively presented in a survey article of Dmitriy Bilyk in the present volume (see [10]). 1…”

“…Accordingly, a lacunary Fourier series is a series which has "gaps" in the sense that it is composed of trigonometric functions whose frequencies are far apart from each other. A classical gap condition is the Hadamard gap condition, requiring that (10)…”

“…for (n k ) k≥1 satisfying (10). By a classical heuristics, lacunary sequences resemble many properties which are typical for sequences of independent random variables.…”