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 \(\pm 1\)) 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.
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Acknowledgments
The authors would like to express gratitude to IMA and NSF: the research of the first author was supported by the NSF grant DMS 1101519, and the second author’s stay at IMA was also supported by NSF funds. The authors are also extremely grateful to Ohad Feldheim for contributing the idea of Sect. 4.3, and to Josef Dick and Wolfgang Schmid for pointing out references [20, 21].
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Communicated by Hans G. Feichtinger.
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Bilyk, D., Feldheim, N. The Two-Dimensional Small Ball Inequality and Binary Nets. J Fourier Anal Appl 23, 817–833 (2017). https://doi.org/10.1007/s00041-016-9491-9
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DOI: https://doi.org/10.1007/s00041-016-9491-9