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Annales Henri Poincaré

, 12:1027 | Cite as

On the Geometry of the Nodal Lines of Eigenfunctions of the Two-Dimensional Torus

  • Jean Bourgain
  • Zeév RudnickEmail author
Article

Abstract

The width of a convex curve in the plane is the minimal distance between a pair of parallel supporting lines of the curve. In this paper we study the width of nodal lines of eigenfunctions of the Laplacian on the standard flat torus. We prove a variety of results on the width, some having stronger versions assuming a conjecture of Cilleruelo and Granville asserting a uniform bound for the number of lattice points on the circle lying in short arcs.

Keywords

Lattice Point Trigonometric Polynomial Nodal Line Total Curvature Convex Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Berry M.: Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature. J. Phys. A 35(13), 3025–3038 (2002)MathSciNetADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Castrillón López M., Fernández Mateos V., Muñoz Masqué J.: Total curvature of curves in Riemannian manifolds. Differ. Geom. Appl. 28(2), 140–147 (2010)CrossRefzbMATHGoogle Scholar
  3. 3.
    Cheng S.Y.: Eigenfunctions and nodal sets. Comment. Math. Helv. 51(1), 43–55 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cilleruelo J., Cordoba A.: Trigonometric polynomials and lattice points. Proc. AMS 115(4), 899–905 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cilleruelo, J., Granville, A.: Lattice points on circles, squares in arithmetic progressions and sumsets of squares, vol. 43, pp. 241–262. CRM Proc. LN, AMS (2007)Google Scholar
  6. 6.
    Cilleruelo J., Granville A.: Close lattice points on circles. Can. J. Math. 61(6), 1214–1238 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Donnelly H., Fefferman C.: Nodal sets of eigenfunctions of Riemannian manifolds. Invent. Math. 93(1), 161–183 (1988)MathSciNetADSCrossRefzbMATHGoogle Scholar
  8. 8.
    Fontes-Merz N.: A multidimensional version of Turán’s lemma. J. Approx. Theory 140(1), 27–30 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jarnik V.: Uber die Gitterpunkte auf konvexen Kurven. Math. Z. 24(1), 500–518 (1926)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Milnor J.W.: On the total curvature of knots. Ann. Math. (2) 52, 248–257 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Nazarov, F.: Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type. Algebra i Analiz 5(4), 3–66 (1993); translation in St. Petersburg Math. J. 5(4), 663–717 (1994)Google Scholar
  12. 12.
    Ramana D.S.: Arithmetical applications of an identity for the Vandermonde determinant. Acta Arith. 130(4), 351–359 (2007)MathSciNetADSCrossRefzbMATHGoogle Scholar
  13. 13.
    Risler J.: On the curvature of the real Milnor fiber. Bull. Lond. Math. Soc. 35, 445–454 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.School of Mathematics, Institute for Advanced StudyPrincetonUSA
  2. 2.Raymond and Beverly Sackler School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

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