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Irregularity of Distribution in Wasserstein Distance

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Abstract

We study the non-uniformity of probability measures on the interval and circle. On the interval, we identify the Wasserstein-p distance with the classical \(L^p\)-discrepancy. We thereby derive sharp estimates in Wasserstein distances for the irregularity of distribution of sequences on the interval and circle. Furthermore, we prove an \(L^p\)-adapted Erdős–Turán inequality, and use it to extend a well-known bound of Pólya and Vinogradov on the equidistribution of quadratic residues in finite fields.

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Acknowledgements

We are indebted to Sarah Peluse, who offered indispensable guidance and references for the analytic number theory in the final section. We also thank Stefan Steinerberger, Andrea Ottolini, and the two anonymous referees for their astute suggestions. This work was supported by the Fannie and John Hertz Foundation and by NSF Grant DGE-1656518.

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  1. Department of Mathematics, Stanford University, 450 Jane Stanford Way, Building 380, Stanford, CA, 94305, USA

    Cole Graham

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  1. Cole Graham
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Correspondence to Cole Graham.

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Communicated by Massimo Fornasier.

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Graham, C. Irregularity of Distribution in Wasserstein Distance. J Fourier Anal Appl 26, 75 (2020). https://doi.org/10.1007/s00041-020-09786-y

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