Nodal intersections of toral eigenfunctions with a curved segment: the Bourgain-Rudnick linear lower-bound conjecture

How many zeros does a Laplace eigenfunction on the flat two-dimensional torus have along a fixed curve? For a curve $C$ with nowhere-vanishing curvature and an eigenfunction $F$ with wavenumber $\lambda$, the number of intersection points of $C$ with the nodal set (zero set) of $F$ is at most of order $\lambda$ when $C$ is real-analytic, and Bourgain and Rudnick (Israel J. Math. 207 (2015) 479-505) proved the lower bound $N_{F,C}\gg\lambda/B_\lambda^{5/2}$, where $B_\lambda$ is the maximal number of lattice points on the frequency circle $|x|=\lambda$ lying in an arc of length about $\sqrt{\lambda}$; unconditionally $N_{F,C}\gg\lambda/(\log\lambda)^{5/2}$. They conjectured the sharp linear law $N_{F,C}\gg_C\lambda$. The conjecture is open as of 2026: no later work proves, refutes, or improves the deterministic lower bound, and the two sufficient routes - the Cilleruelo-Granville lattice-point conjecture $B_\lambda=O(1)$ and the uniform $L^1$ restriction lower bound $\int_C|F|\gg_C\|F\|_2$ - both remain open. This record states the source-faithful problem: establish the linear lower bound uniformly over all eigenfunctions and all fixed curved curves.

Background

Setting and definitions. Let $\mathbb{T}^2=\mathbb{R}^2/2\pi\mathbb{Z}^2$ be the standard flat torus, that is, the square $[0,2\pi)^2$ with opposite edges identified, equipped with the Euclidean Laplacian $\Delta$. A (real-valued) Laplace eigenfunction is a nonzero function $F:\mathbb{T}^2\to\mathbb{R}$ solving $-\Delta F=\lambda^2F$; the parameter $\lambda>0$ is called the wavenumber. On the torus every eigenfunction is a trigonometric polynomial $F(x)=\sum_{\mu\in E_\lambda}c_\mu e^{i\mu\cdot x}$ whose frequencies form the set $E_\lambda$ of integer lattice points on the circle $|x|=\lambda$; nontrivial eigenspaces exist only when $\lambda^2$ is a sum of two squares of integers, and their dimension (the number of such representations) can grow faster than any power of $\log\lambda$. The nodal set of $F$ is its zero curve $\{x:F(x)=0\}$. For a fixed smooth reference curve $C\subset\mathbb{T}^2$ with nowhere-vanishing curvature (no straight pieces), the nodal intersection count is $N_{F,C}=\#\{x\in C:F(x)=0\},$ the number of zeros of $F$ along $C$ - a quantitative diagnostic of how often the nodal line meets a given geometric probe. The notation $X\gg_CY$ means that $X\ge cY$ holds for a constant $c>0$ depending only on the curve $C$, asserted for all sufficiently large wavenumbers, i.e. for all $\lambda\ge\lambda_0(C)$; $O(\cdot)$ and $o(\cdot)$ carry their standard asymptotic meaning.

Known bounds and the arithmetic obstruction. The count is at most linear: $N_{F,C}\ll_C\lambda$ for real-analytic $C$, obtained from a theorem of Toth and Zelditch for analytic curves in analytic domains, applied through the uniform two-sided $L^2$ restriction bounds $\|F\|_{L^2(C)}\asymp_C\|F\|_2$ that Bourgain and Rudnick had earlier established for curved curves on the torus (here $\|F\|_2$ is the $L^2$ norm over the whole torus and $\|F\|_{L^p(C)}=(\int_C|F|^p\,ds)^{1/p}$ are the restriction norms, i.e. the $L^p$ sizes of the eigenfunction restricted to the curve). Bourgain and Rudnick also proved a near-linear lower bound $N_{F,C}\gg\lambda^{1-o(1)}$, and the 2015 source paper sharpened it to an explicit arithmetic form: with $B_\lambda=\max_{|x|=\lambda}\#\{\mu\in E_\lambda:|\mu-x|<\sqrt{\lambda}\}$ the maximal number of frequency-lattice points lying in an arc of length about $\sqrt{\lambda}$ of the circle $|x|=\lambda$, the paper proves $N_{F,C}\gg\lambda/B_\lambda^{5/2}.$ Here $B_\lambda$ is an arithmetic quantity: by a classical theorem of Cilleruelo and Cordoba, arcs of length $\lambda^{\delta}$ with $\delta<1/2$ carry only boundedly many lattice points, and Bourgain and Rudnick proved the unconditional bound $B_\lambda\ll\log\lambda$ for the critical $\sqrt{\lambda}$ arcs (their Lemma 2.1), giving $N_{F,C}\gg\lambda/(\log\lambda)^{5/2}$. The Cilleruelo-Granville conjecture asserts $B_\lambda=O(1)$; it is known for almost all $\lambda$ (Bourgain-Rudnick 2011) but is open in general, and by Theorem 1.1 of the source paper it would imply the linear law.

The restriction-theoretic route. The proof of the lower bound runs through restriction norms. The source paper shows $N_{F,C}\gg\lambda\cdot((1/\|F\|_2)\int_C|F|\,ds)^5$ (its Theorem 1.2), so any uniform lower bound on the $L^1$ restriction norm $\int_C|F|\,ds$ yields a lower bound on the intersection count; by log-convexity of $L^p$ norms, $\|F\|_{L^1(C)}\gg_C\|F\|_2^3/\|F\|_{L^4(C)}^2$, while the $L^4$ restriction norm satisfies $\|F\|_{L^4(C)}\ll B_\lambda^{1/4}\|F\|_2$ (Theorem 1.3), giving $\int_C|F|\,ds\gg_C\|F\|_2/\sqrt{B_\lambda}$. The authors conjecture the uniform $L^1$ restriction lower bound $\int_C|F|\,ds\gg_C\|F\|_2$, which via Theorem 1.2 would also give the linear law.

Why curvature is required. The source paper shows the curvature hypothesis cannot be dropped. If $C$ is a geodesic segment (zero curvature), there are arbitrarily large wavenumbers admitting eigenfunctions with no zeros on $C$ (irrational-slope case) or vanishing identically on $C$ (rational-slope case). On the sphere, an analogue of the lower bound fails even for parallels of positive curvature, because zonal spherical harmonics (the rotationally symmetric ones, given by Legendre polynomials in the colatitude) avoid a fixed parallel for infinitely many degrees, by a classical irreducibility property of Legendre polynomials.

Adjacent literature (status as of September 2026). The average-case picture is understood: for arithmetic random waves - Gaussian models of eigenfunctions with the same frequency sets $E_\lambda$ - the expected nodal intersection count with a segment is universally proportional to $\lambda$, with variance and limiting-distribution results in two and three dimensions (Rudnick-Wigman, Rudnick-Wigman-Yesha, Maffucci, Rossi-Wigman, Maffucci-Rossi). These are probabilistic statements and do not address the uniform deterministic conjecture. On the upper-bound side of restriction theory, Huang-Zhang verified the Bourgain-Rudnick $L^2$ restriction conjecture for rational totally geodesic submanifolds (submanifolds swept out by geodesics of the ambient torus, such as rational-slope affine subtori) and closed geodesics, and Zhang-Zhu (June 2026) proved the Huang-Zhang conjecture in high codimension and new curve estimates on $\mathbb{T}^3$ while explicitly recording that the $\sqrt{\lambda}$-arc lattice-point problem and the uniform $L^2$ bound for geodesics remain open. Wang (August 2026) derived further toral eigenfunction estimates and mixed additive-energy bounds on lattice spheres via bilinear decoupling, again on the upper-bound side. Lower-bound technology in other settings (Toth-Zelditch's geometric control conditions; Canzani-Toth's Carleman estimates - weighted elliptic inequalities of unique-continuation type - in lacunary regions) yields only exponentially small bounds under dynamical hypotheses. A 2021 preprint of Oganesyan claiming to disprove a Cilleruelo-Granville-type uniform bound for arcs of length $R^{\alpha}$, $\alpha>1/2$, was withdrawn due to a mistake. As of September 2026 the deterministic linear lower-bound conjecture, the uniform $L^1$ restriction conjecture, and the Cilleruelo-Granville conjecture all remain open.

Problem Statement

Prove or refute conjecture (1.5) of Bourgain-Rudnick (Israel J. Math. 207 (2015) 479-505), an asymptotic statement as the wavenumber $\lambda\to\infty$: for every fixed smooth curve $C\subset\mathbb{T}^2=\mathbb{R}^2/2\pi\mathbb{Z}^2$ with nowhere-vanishing curvature there exist a constant $c_C>0$ and a threshold $\lambda_0=\lambda_0(C)$ such that, for every admissible wavenumber $\lambda\ge\lambda_0$ (that is, every $\lambda\ge\lambda_0$ whose eigenspace is nonzero) and every real-valued Laplace eigenfunction $F$ on the torus with wavenumber $\lambda$ (i.e. $-\Delta F=\lambda^2F$), the nodal intersection count $N_{F,C}=\#\{x\in C:F(x)=0\}$ satisfies $N_{F,C}\ge c_C\lambda$; equivalently, $N_{F,C}/\lambda$ stays bounded below by a positive constant depending only on $C$ as $\lambda\to\infty$ along admissible wavenumbers. The claim is asymptotic in $\lambda$: a single wavenumber, or a single eigenfunction, never resolves it, and a resolution must hold for every sufficiently large admissible $\lambda$.

The conjecture is sharp: $N_{F,C}\ll_C\lambda$ is known for real-analytic $C$, while the best unconditional lower bound is $N_{F,C}\gg\lambda/B_\lambda^{5/2}\ge\lambda/(\log\lambda)^{5/2}$, where $B_\lambda$ is the maximal number of integer lattice points on the circle $|x|=\lambda$ lying in an arc of length about $\sqrt{\lambda}$. The source paper identifies two sufficient conditions, each open: (i) the Cilleruelo-Granville conjecture $B_\lambda=O(1)$, which combined with its Theorem 1.1 gives the linear law; (ii) the uniform $L^1$ restriction lower bound $\int_C|F|\,ds\gg_C\|F\|_{L^2(\mathbb{T}^2)}$, which combined with its Theorem 1.2 gives the linear law. The statement must remain uniform over all eigenfunctions of all admissible wavenumbers $\lambda\ge\lambda_0$ and all fixed curves with nowhere-vanishing curvature: no density-one-subsequence, generic-wavenumber, model-class, or restricted-geometry weakenings qualify, and the curvature hypothesis itself is part of the source statement (for geodesic segments the conclusion is false).

References

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