Irregularity of Distribution in Wasserstein Distance
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(2020) 26:75
Irregularity of Distribution in Wasserstein Distance Cole Graham1 Received: 14 November 2019 / Revised: 24 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 Erd˝os–Turán inequality, and use it to extend a well-known bound of Pólya and Vinogradov on the equidistribution of quadratic residues in finite fields. Keywords Irregularity of distribution · Optimal transport · Wasserstein distance Mathematics Subject Classification 11K38 · 11K06 · 42A05
1 Introduction We consider the classical question of irregularity of distribution: if we successively place points in a box, how evenly can we space them? Answers encompass a vast body of theoretical and numerical work. Rather than cite all related literature, we direct the reader to the excellent survey [3] and monograph [1]. In this note, we restrict our attention to one dimension. Given a sequence of points (xn )n∈N in the interval I = [0, 1) how well can the empirical measures
μ N :=
N 1 δxn N
(1)
n=1
Communicated by Massimo Fornasier.
B 1
Cole Graham [email protected] Department of Mathematics, Stanford University, 450 Jane Stanford Way, Building 380, Stanford, CA 94305, USA 0123456789().: V,-vol
75
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Journal of Fourier Analysis and Applications
(2020) 26:75
approximate the uniform distribution λ? Our answer involves several distances d on the space of probability measures P(I ), but all agree that if the first N points are evenly spaced on I , d(μ N , λ) ∼
C . N
(2)
N However, the truncations (xn )n=1 of a fixed sequence (xn )n∈N cannot all be evenly spaced, so we naturally wonder whether (2) can hold for all N ∈ N. Indeed, van der Corput conjectured and van Aardenne-Ehrenfest confirmed that such uniform even spacing is impossible [33,34]. These classical results concern distances d based on the discrepancy function
D N (x) := μ N ([0, x)) − x for x ∈ I . For p ∈ [1, ∞], the L p -discrepancy of a sequence (xn ) at stage N is D N L p . Thus the L ∞ -discrepancy is simply the Kolmogorov–Smirnov statistic [18]. The following theorem sharply answers van der Corput’s conjecture in L p -discrepancy, and unites the work of numerous authors. To state it cleanly, we let α p :=
if p ∈ [1, ∞), 1 if p = ∞. 1 2
Theorem 1 [5,6,14,19,27,29,35] For every p ∈ [1, ∞], there exists a constant C p > 0 such that for any sequence (xn )n∈N ⊂ I , D N L p ≥ C p
logα p N N
holds for infinitely many N ∈ N. Furthermore, this bound is sharp. Remark 1 Since D N is an antiderivative of μ N − λ, we in fact have D N L p = μ N − λW˙ −1, p (I ) .
(3)
Thus Theorem 1 quantifies the non-uniformity of μ N in the negative Sobolev norm · W˙ −1, p (I ) , which
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