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

Jean Bourgain; Zeév Rudnick

doi:10.1007/s00023-011-0098-z

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

Jean Bourgain
, Zeév Rudnick

School of Mathematical Sciences

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	1027-1053
Number of pages	27
Journal	Annales Henri Poincare
Volume	12
Issue number	6
DOIs	https://doi.org/10.1007/s00023-011-0098-z
State	Published - Sep 2011

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Nuclear and High Energy Physics
Mathematical Physics

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10.1007/s00023-011-0098-z

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title = "On the Geometry of the Nodal Lines of Eigenfunctions of the Two-Dimensional Torus",

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.",

author = "Jean Bourgain and Ze{\'e}v Rudnick",

note = "Funding Information: We thank Misha Sodin for his comments. J.B. was supported in part by N.S.F. grant DMS 0808042. Z.R. was supported by the Oswald Veblen Fund during his stay at the Institute for Advanced Study and by the Israel Science Foundation (grant No. 1083/10).",

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AU - Rudnick, Zeév

N1 - Funding Information: We thank Misha Sodin for his comments. J.B. was supported in part by N.S.F. grant DMS 0808042. Z.R. was supported by the Oswald Veblen Fund during his stay at the Institute for Advanced Study and by the Israel Science Foundation (grant No. 1083/10).

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N2 - 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.

AB - 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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U2 - 10.1007/s00023-011-0098-z

DO - 10.1007/s00023-011-0098-z

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JO - Annales Henri Poincare

JF - Annales Henri Poincare

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ER -

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

Abstract

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