The Two-Dimensional Small Ball Inequality and Binary Nets

Dmitriy Bilyk, Naomi Feldheim

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

In the current paper we present a new proof of the small ball inequality in two dimensions. More importantly, this new argument, based on an approach inspired by lacunary Fourier series, reveals the first formal connection between this inequality and discrepancy theory, namely the construction of two-dimensional binary nets, i.e. finite sets which are perfectly distributed with respect to dyadic rectangles. This relation allows one to generate all possible point distributions of this type. In addition, we outline a potential approach to the higher-dimensional small ball inequality by a dimension reduction argument. In particular this gives yet another proof of the two-dimensional signed (i.e. coefficients ± 1) small ball inequality by reducing it to a simple one-dimensional estimate. However, we show that an analogous estimate fails to hold for arbitrary coefficients.

Original languageEnglish (US)
Pages (from-to)817-833
Number of pages17
JournalJournal of Fourier Analysis and Applications
Volume23
Issue number4
DOIs
StatePublished - Aug 1 2017

Bibliographical note

Publisher Copyright:
© 2016, Springer Science+Business Media New York.

Keywords

  • Digital nets
  • Discrepancy
  • Haar functions
  • Small ball conjecture

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