Width of regular nodal arcs and total curvature of nodal sets for eigenfunctions of the two-dimensional flat torus (the Bourgain–Rudnick open problems)

Bourgain and Rudnick (Annales Henri Poincaré 12, 2011) proposed measuring the fine geometry of nodal lines of Laplace eigenfunctions — for which pointwise curvature is not a usable notion — by the width of regular arcs of the nodal set and by the total curvature of the nodal set, and left several open questions which remain open to date. The central one, their Conjecture 1, asserts that for every $\varepsilon>0$ there is $c_\varepsilon>0$ such that every regular arc $C$ contained in the nodal set of every eigenfunction of the flat torus with eigenvalue $E=\lambda^2$ satisfies $w(C)<c_\varepsilon\lambda^{-1+\varepsilon}$; unconditionally one knows only $w(C)\ll_\varepsilon\lambda^{-1/3+\varepsilon}$ for all arcs, negligibility of the total length of arcs of width exceeding $\lambda^{-1/2+\varepsilon}$, and the full bound for a density-one set of eigenvalues. The number-theoretic route through the Cilleruelo–Granville conjecture on lattice points in short arcs of circles remains open (a 2021 claimed disproof was withdrawn), and the companion question — whether the total curvature bound $K_\varphi\ll E$, proved on the torus and the sphere, extends to general real-analytic compact surfaces, or whether any explicit bound $K(E)$ exists at all — is likewise untouched.

Background

On the standard flat two-dimensional torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$, the eigenvalues of the Laplacian are $4\pi^2E$ where $E=n_1^2+n_2^2$ is an integer that is a sum of two squares; following Bourgain–Rudnick one writes $E=\lambda^2$ and (by a standard abuse) calls $E$ the eigenvalue. The corresponding eigenspace consists of trigonometric polynomials $\varphi(x)=\sum_{\xi\in\mathbb{Z}^2,\,|\xi|^2=E}a_\xi e(x\cdot\xi)$, where $e(t)=e^{2\pi i t}$ and $a_\xi=a_{-\xi}$ for real-valued $\varphi$. The nodal set $\mathcal{N}_\varphi=\{\varphi=0\}$ is, apart from finitely many singular points at which nodal arcs meet in equiangular systems, a union of $C^2$-immersed closed curves (nodal lines). By the Donnelly–Fefferman bounds, on any real-analytic surface the length of the nodal set of an eigenfunction with $E=\lambda^2$ is commensurable to the wavelength scale, $\mathrm{length}(\mathcal{N}_\varphi)\approx\lambda$; this says nothing about how the nodal lines bend.

Berry's random-wave model for high-frequency chaotic eigenfunctions predicts that typical nodal lines bend on the wavelength scale. For individual, deterministic eigenfunctions pointwise curvature is not a usable notion: nodal lines may contain straight segments (for $\sin nx$ the nodal set is a union of geodesics) and may also develop arbitrarily large pointwise curvature at a fixed eigenvalue. Bourgain and Rudnick therefore proposed two global substitutes for curvature.

The first is the width of regular arcs. An arc $C\subset\mathbb{T}^2$ is called regular if it admits an arc-length parametrization $\gamma:[0,\ell]\to C$ with $0<\ell<1$ that is $C^2$ and whose curvature is pinched, $\kappa<|\ddot\gamma|<2\kappa$ for some $\kappa>0$, with total curvature $2\kappa\ell<1$. For a convex curve the width $w(C)$ is the minimal distance between a pair of parallel supporting lines; for a regular arc $w(C)\approx\ell^2\kappa$, so the width measures how much room a curved nodal arc needs. Numerics suggest that nodal sets contain no large curved arcs: as $E\to\infty$, every regular arc of the nodal set must either shrink ($\ell\to0$) or flatten ($\kappa\to0$).

The second is the total curvature. For a $C^2$ curve the total curvature is $\int\|\ddot\gamma(s)\|\,ds$; in Milnor's definition it extends to any continuous curve as the supremum of the total curvatures of inscribed polygons, a quantity well defined on any Riemannian surface. On the flat torus the total curvature $K_\varphi$ of the nodal set has no lower bound (it vanishes for $\sin nx$); the question is an upper bound in terms of the eigenvalue.

The proofs depend on the arithmetic of the frequency circle: the set $\mathcal{E}=\mathbb{Z}^2\cap\{|x|=\lambda\}$ of lattice points on the circle of radius $\lambda$ has size $r_2(E)$ — the number of representations of $E$ as an ordered sum of two squares — which can be as large as $\exp(c\log E/\log\log E)$. Classically, Jarník (1926) proved that any arc of length $\ll\lambda^{1/3}$ on this circle contains at most two lattice points, and Cilleruelo–Córdoba (1992) proved that for every $\delta<1/2$ there is $M(\delta)$ such that any arc of length $\lambda^\delta$ contains at most $M(\delta)$ lattice points. The Cilleruelo–Granville conjecture (2007) asserts the same up to any exponent below one: for each $\varepsilon\in(0,1)$ there is $B_\varepsilon$ such that any arc of length $<\lambda^{1-\varepsilon}$ on $\{|x|=\lambda\}$ contains at most $B_\varepsilon$ lattice points, uniformly in $\lambda$.

With these notions Bourgain–Rudnick proved: (Theorem 1) every regular nodal arc satisfies $w(C)<C_\varepsilon\lambda^{-1/3+\varepsilon}$; (Theorem 2) the collection of disjoint regular arcs with $w(C_\alpha)>\lambda^{-1/2+\varepsilon}$ has total length $\ll\lambda^{1-\delta}$, a negligible part of the nodal set whose total length is $\approx\lambda$; (Theorem 3) after deleting all arcs of width $>\lambda^{-1+\varepsilon}$, a positive proportion (more than $c_0\lambda$) of the nodal length remains; and (Section 4.2) the full Conjecture-1 bound holds for a density-one set of eigenvalues, namely those $E$ for which the lattice points on the frequency circle are $\lambda^{1-\varepsilon}$-separated, which holds for all but $O(N^{1-\varepsilon/3})$ integers $E\le N$. Under the Cilleruelo–Granville conjecture, the exponent $1/2$ in Theorem 2 can be replaced by $1$, matching the conjectured scale. For the total curvature they showed $K_\varphi\ll E$ on the torus — and, by the same argument, on the sphere — by writing $\varphi$ as an algebraic function of $u_i=\cos x_i$ and applying the known bound on the curvature of real algebraic curves of degree $d<c\lambda$.

Two caveats recorded in the source: the width statements are specific to the flat torus and fail on the sphere $S^2$, where the spherical harmonics $Y_{\ell,m}=P_{\ell,m}(\theta)e^{im\varphi}$ have latitude circles $\{P_{\ell,m}(\theta)=0\}$ as families of regular arcs with geodesic curvature (the curvature of the curve as measured within the surface) bounded away from zero; and higher-order smoothness of regular arcs does not hold automatically — an explicit trigonometric example in Appendix A of the source has a convex nodal arc of curvature $\sim1/k$ but fourth derivative $\sim k$ — so any route to better exponents via higher derivatives must impose such assumptions artificially.

Problem Statement

The problem consists of the open questions left by Bourgain–Rudnick (2011) on the fine geometry of nodal lines; they share one scientific objective — deterministic, quantitative control of the local geometry of nodal sets of Laplace eigenfunctions beyond their total length. It has a primary and a companion component, each of which is posed as a prove-or-disprove question.

Component 1 (primary — the width conjecture). Prove or disprove Conjecture 1 of the source paper: for every $\varepsilon>0$ there exists $c_\varepsilon>0$ such that for every eigenfunction $\varphi$ of the flat torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ with eigenvalue $E=\lambda^2$ and every regular arc $C$ contained in the nodal set $\mathcal{N}_\varphi$ (regular in the sense of the source: $C$ admits an arc-length parametrization $\gamma:[0,\ell]\to C$ with $0<\ell<1$ whose curvature is pinched, $\kappa<|\ddot\gamma|<2\kappa$, and $2\kappa\ell<1$), the width satisfies

$$w(C)<c_\varepsilon\lambda^{-1+\varepsilon}.$$

A refutation must exhibit, for some fixed $\varepsilon_0>0$, an explicit infinite family of eigenfunctions $\varphi_j$ with $\lambda_j\to\infty$ and regular arcs $C_j\subseteq\mathcal{N}_{\varphi_j}$ whose widths satisfy $w(C_j)\gg\lambda_j^{-1+\varepsilon_0}$, i.e. $w(C_j)\lambda_j^{1-\varepsilon_0}\to\infty$ as $j\to\infty$. Two intermediate questions, each of which is precisely formulated but neither of which by itself proves or disproves Conjecture 1: (i) improve the unconditional exponent $1/3$ of Theorem 1, i.e. prove $w(C)\ll_\varepsilon\lambda^{-\sigma+\varepsilon}$ for some $\sigma>1/3$ for all eigenfunctions and all regular arcs; (ii) resolve the Cilleruelo–Granville conjecture (Conjecture 2 of the source): for every $\varepsilon\in(0,1)$ there is $B_\varepsilon<\infty$ such that every arc of length $<\lambda^{1-\varepsilon}$ on the circle $\{x\in\mathbb{R}^2:|x|=\lambda\}$ contains at most $B_\varepsilon$ points of $\mathbb{Z}^2$; a positive answer upgrades the exponent $1/2$ in Theorem 2 to $1$, while a refutation must exhibit radii $\lambda_j\to\infty$ and arcs of length $\le\lambda_j^{1-\varepsilon}$ containing unboundedly many lattice points.

Component 2 (companion — total curvature). Determine whether the total curvature bound $K_\varphi\ll E$ — established for the flat torus and, by the same argument, for the sphere — extends to the nodal sets of eigenfunctions on general real-analytic compact surfaces, or, more modestly, whether any explicit bound $K_\varphi<K(E)$ holds; a negative answer must exhibit an explicit infinite family of real-analytic compact surfaces with eigenfunctions $\varphi_j$ of eigenvalue $E_j\to\infty$ such that $K_{\varphi_j}\gg E_j^{m}$ for some fixed $m>1$, so that the strong bound $K_\varphi\ll E$ fails. Here $K_\varphi$ is the total curvature of the nodal set in Milnor's sense, the supremum of the total curvatures of inscribed polygons. A proof of $K_\varphi\ll E$ on all real-analytic compact surfaces, or an explicit bound $K(E)$ depending only on the eigenvalue and valid for every real-analytic compact surface, completes this component affirmatively; a family as in the negative branch completes it by refutation, answering negatively the question whether $K_\varphi\ll E$ extends to general real-analytic compact surfaces and leaving open whether some weaker explicit bound $K(E)$ still holds.

The width statement does not extend beyond the flat torus: the source shows that Conjecture 1 fails on the sphere $S^2$ and, in dimensions $\ge3$, that arbitrary hypersurfaces can occur as nodal components, so the width phenomenon, if true, is specific to the two-dimensional arithmetic setting. This concerns the width component only; the total-curvature question of Component 2 on general real-analytic compact surfaces remains part of the required answer.

References

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