Advertisement

Geometric and Functional Analysis

, Volume 28, Issue 4, pp 1131–1181 | Cite as

A discretized Severi-type theorem with applications to harmonic analysis

  • Joshua ZahlEmail author
Article

Abstract

In 1901, Severi proved that if \({{Z}}\) is an irreducible hypersurface in \({\mathbb{P}^{4}(C)}\) that contains a three dimensional set of lines, then \({{Z}}\) is either a quadratic hypersurface or a scroll of planes. We prove a discretized version of this result for hypersurfaces in \({\mathbb{R}^{4}}\). As an application, we prove that at most \({\delta^{{-2}{-\epsilon}}}\) direction-separated δ-tubes can be contained in the δ-neighborhood of a low-degree hypersurface in \({\mathbb{R}^{4}}\). This result leads to improved bounds on the restriction and Kakeya problems in \({\mathbb{R}^{4}}\). Combined with previous work of Guth and the author, this result implies a Kakeya maximal function estimate at dimension 3+1/28, which is an improvement over the previous bound of 3 due to Wolff. As a consequence, we prove that every Besicovitch set in \({\mathbb{R}^{4}}\) must have Hausdorff dimension at least 3 + 1/28. Recently, Demeter showed that any improvement over Wolff’s bound for the Kakeya maximal function yields new bounds on the restriction problem for the paraboloid in \({\mathbb{R}^{4}}\).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BB16.
    Barone S., Basu. S.: On a real analog of Bezout inequality and the number of connected components of sign conditions. Proc. London Math. Soc., (1) 112, 115–145 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. BCR98.
    Bochnak J., Coste M., Roy M.: Real algebraic geometry. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  3. BG73.
    Brudnyi Y., Ganzburg I.: On an extremal problem for polynomials in n variables. Math. USSR Izvestijia 7, 345–356 (1973)CrossRefGoogle Scholar
  4. Cor77.
    Córdoba A.: The Kakeya maximal function and the spherical summation multiplier. Am. J. Math. 99, 1–22 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Dem17.
    C. Demeter. On the restriction theorem for paraboloid in \({\mathbb{R}^{4}}\). arXiv:1701.03523. (2017).
  6. Dvi09.
    Dvir Z.: On the size of Kakeya sets in finite fields. J. Amer. Math. Soc. 22, 1093–1097 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. DKSS13.
    Dvir Z., Kopparty S., Saraf S., Sudan M.: Extensions to the method of multiplicities, with applications to Kakeya sets and mergers. SIAM J. Comput.(6) 42, 2305–2328 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Fef70.
    Fefferman C.: Inequalities for strongly singular convolution operators. Acta Math. 124, 9–36 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Gut16a.
    Guth L.: Restriction estimates using polynomial partitioning. J. Amer. Math. Soc. (2) 29, 371–413 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Gut16b.
    L. Guth. Restriction estimates using polynomial partitioning II. arXiv:1603.04250. (2016).
  11. GZ17.
    L. Guth, J. Zahl. Polynomial Wolff axioms and Kakeya-type estimates in \({\mathbb{R}^{4}}\). To appear, Proc. London Math. Soc., arXiv:1701.07045. (2017).
  12. KLT00.
    Katz N., Łaba I., Tao T.: An improved bound on the Minkowski dimension of Besicovitch sets in \({\mathbb{R}^{3}}\). Ann. of Math. 152, 383–446 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. KR18.
    N. Katz, K. Rogers. On the polynomial Wolff axioms. arXiv:1802.09094. (2018).
  14. KZ17.
    N. Katz, J. Zahl. An improved bound on the Hausdorff dimension of Besicovitch sets in \({\mathbb{R}^{3}}\). arXiv:1704.07210. (2017).
  15. LT01.
    Łaba I., Tao T.: An improved bound for the Minkowski dimension of Besicovitch sets in medium dimension. Geom. Funct. Anal. (4) 11, 773–806 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Mil64.
    Milnor J.: On the Betti numbers of real varieties. Proc. Amer. Math. Soc. (2) 15, 275–280 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Rog94.
    Rogora E.: Varieties with many lines. Manuscripta Math.(1) 82, 207–226 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Seg48.
    B. Segre. Sulle \({{V}_{n}}\) aventi piú di \(\infty^{n-k}{S}_{k}\). Rendiconti dellaccademia nazionale dei Lincei., Vol. V, note I e II. (1948).Google Scholar
  19. Sev01.
    Severi F.: Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni, e a’suoi punti tripli apparenti. Rend. circ. mat. Palermo, (1) 15, 33–51 (1901)CrossRefzbMATHGoogle Scholar
  20. SS17.
    Sharir M., Solomon. N.: Incidences between points and lines in \({\mathbb{R}^{4}}\). Disc. Comput. Geom. (3) 57, 702–756 (2017)CrossRefzbMATHGoogle Scholar
  21. Tao05.
    Tao T.: A new bound for finite field Besicovitch sets in four dimensions. Pacific J. Math. 222, 43–57 (2005)MathSciNetCrossRefGoogle Scholar
  22. Wol95.
    Wolff T.: An improved bound for Kakeya type maximal functions. Rev. Mat. Iberoam. (3) 11, 651–674 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Won03.
    Wongkew R.: Volumes of tubular neighbourhoods of real algebraic varieties. Pacific J Math. 159, 177–184 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. YC04.
    Yomdin Y., Comte G.: Tame geometry with application in smooth analysis. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  25. Zyg74.
    Zygmund A.: On Fourier coefficients and transforms of functions of two variables. Studia Math. 50, 189–201 (1974)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of British ColumbiaVancouverCanada

Personalised recommendations