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Irregularity of Distribution in Wasserstein Distance

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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References

  1. Beck, J., Chen, W.W.L.: Irregularities of Distribution. Cambridge Tracts in Mathematics, vol. 89. Cambridge University Press, Cambridge (1987)

    Book  Google Scholar 

  2. Benamou, J.D., Brenier, Y.: A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84(3), 375–393 (2000)

    MathSciNet  Article  Google Scholar 

  3. Bilyk, D.: On Roth’s orthogonal function method in discrepancy theory. Unif. Distrib. Theory 6(1), 143–184 (2011)

    MathSciNet  MATH  Google Scholar 

  4. Brown, L., Steinerberger, S.: On the Wasserstein distance between classical sequences and the Lebesgue measure. Trans. Am. Math. Soc. (2020)

  5. Chen, W.W.L.: On irregularities of distribution. Mathematika 27(2), 153–170 (1980)

    MathSciNet  Article  Google Scholar 

  6. Davenport, H.: Note on irregularities of distribution. Mathematika 3(2), 131–135 (1956)

    MathSciNet  Article  Google Scholar 

  7. Davenport, H.: Multiplicative Number Theory. Graduate Texts in Mathematics, vol. 74, 3rd edn. Springer, New York (2000). Revised and with a preface by Hugh L. Montgomery

    MATH  Google Scholar 

  8. Delon, J., Salomon, J., Sobolevski, A.: Fast transport optimization for Monge costs on the circle. SIAM J. Appl. Math. 70(7), 2239–2258 (2010)

    MathSciNet  Article  Google Scholar 

  9. Erdős, P., Turán, P.: On a problem in the theory of uniform distribution I. Nederl. Akad. Wetensch. Proc. 51, 1146–1154 (1948)

    MathSciNet  MATH  Google Scholar 

  10. Erdős, P., Turán, P.: On a problem in the theory of uniform distribution II. Nederl. Akad. Wetensch. Proc. 51, 1262–1269 (1948)

    MathSciNet  MATH  Google Scholar 

  11. Ganelius, T.: Some applications of a lemma on Fourier series. Publ. Inst. Math. 11(17), 9–18 (1957)

    MathSciNet  MATH  Google Scholar 

  12. Givens, C.R., Shortt, R.M.: A class of Wasserstein metrics for probability distributions. Michigan Math. J. 31(2), 231–240 (1984)

    MathSciNet  Article  Google Scholar 

  13. Granville, A., Soundararajan, K.: The distribution of values of \(L(1,\chi _d)\). Geom. Funct. Anal. 13(5), 992–1028 (2003). https://doi.org/10.1007/s00039-003-0438-3

    MathSciNet  Article  MATH  Google Scholar 

  14. Halász, G.: On Roth’s method in the theory of irregularities of point distributions. Recent Prog. Anal. Number Theory 2, 79–94 (1981)

    MathSciNet  MATH  Google Scholar 

  15. Kantorovič, L.V., Rubinšteĭn, G.Š.: On a space of completely additive functions. Vestnik Leningrad Univ. 13(7), 52–59 (1958)

    MathSciNet  Google Scholar 

  16. Koksma, J.F.: A general theorem from the theory of uniform distribution modulo 1. Math. Zutphen. B 11, 7–11 (1942)

    MathSciNet  Google Scholar 

  17. Koksma, J.F.: Some integrals in the theory of uniform distribution modulo 1. Math. Zutphen. B 11, 49–52 (1942)

    MathSciNet  MATH  Google Scholar 

  18. Kolmogorov, A.: Sulla determinazione empirica di una legge di distribuzione. Giorn. Ist. Ital. Attuari 4, 83–91 (1933)

    MATH  Google Scholar 

  19. Lerch, M.: Question 1547. L’Intermed. Math. 11, 144–145 (1904)

    Google Scholar 

  20. Montgomery, H.L., Vaughan, R.C.: Exponential sums with multiplicative coefficients. Invent. Math. 43(1), 69–82 (1977). https://doi.org/10.1007/BF01390204

    MathSciNet  Article  MATH  Google Scholar 

  21. Otto, F., Villani, C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173(2), 361–400 (2000). https://doi.org/10.1006/jfan.1999.3557

    MathSciNet  Article  MATH  Google Scholar 

  22. Paley, R.E.A.C.: A theorem on characters. J. Lond. Math. Soc. 7(1), 28–32 (1932). https://doi.org/10.1112/jlms/s1-7.1.28

    MathSciNet  Article  MATH  Google Scholar 

  23. Peyre, R.: Comparison between \({W}_2\) distance and \(\dot{H}^{-1}\) norm, and localisation of Wasserstein distance. ArXiv e-prints arXiv:1104.4631 (2011)

  24. Pólya, G.: Über die Verteilung der quadratischen Reste und Nichtreste, pp. 21–29. Göttingen, Nachrichten (1918)

    MATH  Google Scholar 

  25. Proĭnov, P.D.: On irregularities of distribution. C. R. Acad. Bulgare Sci. 39(9), 31–34 (1986)

    MathSciNet  Google Scholar 

  26. Proĭnov, P.D., Grozdanov, V.S.: On the diaphony of the van der Corput–Halton sequence. J. Number Theory 30(1), 94–104 (1988). https://doi.org/10.1016/0022-314X(88)90028-5

    MathSciNet  Article  MATH  Google Scholar 

  27. Roth, K.F.: On irregularities of distribution. Mathematika 1(2), 73–79 (1954)

    MathSciNet  Article  Google Scholar 

  28. Santambrogio, F.: Optimal Transport for Applied Mathematicians. Progress in Nonlinear Differential Equations and their Applications, vol. 87. Birkhäuser, Cham (2015)

    Book  Google Scholar 

  29. Schmidt, W.M.: Irregularities of distribution VII. Acta Arith. 21, 45–50 (1972)

    MathSciNet  Article  Google Scholar 

  30. Steinerberger, S.: Wasserstein distance, Fourier series and applications. ArXiv e-prints arXiv:1803.08011 (2018)

  31. Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat. 36(2), 423–439 (1965)

    MathSciNet  Article  Google Scholar 

  32. Vallender, S.S.: Calculation of the Wasserstein distance between probability distributions on the line. Theory Probab. Appl. 18(4), 784–786 (1974)

    Article  Google Scholar 

  33. van Aardenne-Ehrenfest, T.: Proof of the impossibility of a just distribution of an infinite sequence of points over an interval. Nederl. Akad. Wetensch. Proc. 48, 266–271 (1945)

    MathSciNet  MATH  Google Scholar 

  34. van der Corput, J.G.: Verteilungsfunktionen I. Nederl. Akad. Wetensch. Proc. 38, 813–821 (1935)

    MATH  Google Scholar 

  35. van der Corput, J.G.: Verteilungsfunktionen II. Nederl. Akad. Wetensch. Proc. 38, 1058–1068 (1935)

    MATH  Google Scholar 

  36. Villani, C.: Topics in optimal transportation, vol. 58. American Mathematical Society, Providence (2003)

    MATH  Google Scholar 

  37. Vinogradov, I.M.: Über die Verteilung der quadratischen Reste und Nichtreste. J. Soc. Phys. Math. Univ. Permi 2, 1–14 (1919)

    Google Scholar 

  38. Zinterhof, P.: Über einige Abschätzungen bei der Approximation von Funktionen mit Gleichverteilungsmethoden. Sitzungsber. Österr. Akad. Wiss. Math.-Naturwiss. Kl. II 185(1–3), 121–132 (1976)

    MATH  Google Scholar 

  39. Zinterhof, P., Stegbuchner, H.: Trigonometrische Approximation mit Gleichverteilungsmethoden. Studia Sci. Math. Hung. 13(3–4), 273–289 (1978)

    MathSciNet  MATH  Google Scholar 

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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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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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  • DOI: https://doi.org/10.1007/s00041-020-09786-y

Keywords

  • Irregularity of distribution
  • Optimal transport
  • Wasserstein distance

Mathematics Subject Classification

  • 11K38
  • 11K06
  • 42A05