Annales Henri Poincaré

, 12:1027

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


  • Jean Bourgain
    • School of Mathematics, Institute for Advanced Study
    • Raymond and Beverly Sackler School of Mathematical SciencesTel Aviv University

DOI: 10.1007/s00023-011-0098-z

Cite this article as:
Bourgain, J. & Rudnick, Z. Ann. Henri Poincaré (2011) 12: 1027. doi:10.1007/s00023-011-0098-z


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.

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© Springer Basel AG 2011