Annales Henri Poincaré

, Volume 12, Issue 6, pp 1027-1053

First online:

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

  • Jean BourgainAffiliated withSchool of Mathematics, Institute for Advanced Study
  • , Zeév RudnickAffiliated withRaymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University Email author 


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.