Sharp $L^p$ bounds for Laplace eigenfunctions on the square torus (Bourgain's discrete restriction conjecture)

How large can the $L^p$ norm of an $L^2$-normalized Laplace eigenfunction on the flat torus $\mathbb{T}^d$ be, as a function of the eigenvalue? Bourgain's 1993 discrete restriction conjecture predicts that $M_{d,p}(N)$, the supremum of $\|e_N\|_{L^p}/\|e_N\|_{L^2}$ over eigenfunctions with $\sqrt{-\Delta}$-eigenvalue $2\pi N$, satisfies $M_{d,p}(N)\lesssim_\epsilon N^\epsilon(N^{\frac{d-2}{2}-\frac{d}{p}}+1)$, with the $N^\epsilon$ loss conjectured unnecessary for $d\ge5$. After three decades of progress -- $\ell^2$ decoupling proves the conjecture for $p\le\frac{2(d+1)}{d-1}$, circle-method arguments (with decoupling) for $p\ge\frac{2(d-1)}{d-3}$, and sharp loss-free bounds are known for $d\ge5$, $p>\frac{2d}{d-4}$ -- the conjecture remains open around its critical exponent $p_c=\frac{2d}{d-2}$ in every dimension $d\ge3$, and in $d=2$ even the uniform boundedness of $M_{2,p}(N)$ for a single finite $p>4$ is open.

Background

The flat (square) torus is $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$, the unit cube with opposite faces identified; it is the simplest compact manifold carrying an arithmetic structure. Writing $e(x)=e^{2\pi ix}$, every $f\in L^2(\mathbb{T}^d)$ has a Fourier series $f(x)=\sum_{k\in\mathbb{Z}^d}\mathcal{F}(f)(k)e(k\cdot x)$, and each plane wave $e(k\cdot x)$ is an eigenfunction of the (positive) Laplacian $-\Delta$ with eigenvalue $4\pi^2|k|^2$. An eigenfunction of $\sqrt{-\Delta}$ with eigenvalue $2\pi N$ is therefore a trigonometric polynomial whose frequencies all lie on the sphere of radius $N$ in the integer lattice: $e_N(x)=\sum_{|\xi|=N}a_\xi e(\xi\cdot x)$; the corresponding $-\Delta$ eigenvalue is $4\pi^2N^2$, and one writes $\lambda=N^2$. The quantity of interest is $M_{d,p}(N)=\sup\|e_N\|_{L^p(\mathbb{T}^d)}/\|e_N\|_{L^2(\mathbb{T}^d)}$, the supremum over all nonzero such eigenfunctions, i.e. the operator norm of the spectral projection onto one eigenspace. Because these eigenspaces are highly degenerate (their dimension is the number $\#\mathcal{F}_{d,N}$ of lattice points $\xi\in\mathbb{Z}^d$ with $|\xi|=N$), $M_{d,p}(N)$ can grow with $N$, and determining its exact power growth is a meeting point of harmonic analysis and number theory. Two extremal examples set the conjectured scale. Aligning all phases at one point (all $a_\xi$ equal) gives $M_{d,p}(N)\gtrsim(\#\mathcal{F}_{d,N})^{\frac12}N^{-d/p}$, which is $\gtrsim N^{\frac{d-2}{2}-\frac{d}{p}}$ for $d\ge5$, where $\#\mathcal{F}_{d,N}\sim N^{d-2}$ by the Siegel mass formula (an exact asymptotic formula, from the arithmetic theory of quadratic forms, for the number of lattice points on a sphere); for $d=2,3,4$ only the weaker bound $\#\mathcal{F}_{d,N}\lesssim_\epsilon N^{d-2+\epsilon}$ holds, and it is sharp for infinitely many $N$. A single plane wave gives $M_{d,p}\ge1$. Bourgain's discrete restriction conjecture asserts that these examples are essentially the whole story: for all $d\ge2$, $p\ge2$, $\epsilon>0$, $M_{d,p}(N)\lesssim_\epsilon N^\epsilon\Big(N^{\frac{d-2}{2}-\frac{d}{p}}+1\Big),$ with the $N^\epsilon$ loss expected to be removable when $d\ge5$, and expected to be necessary for large $p$ when $d\le4$, reflecting the arithmetic irregularity of lattice point counts in low dimension. The two terms balance at the critical exponent $p_c=\frac{2d}{d-2}$, where the conjecture becomes $M_{d,p_c}(N)\lesssim_\epsilon N^\epsilon$: no power growth at all. This should be compared with Sogge's sharp cluster estimates for a general compact Riemannian manifold -- bounds on the $L^p$ norms of spectral clusters, i.e. sums of eigenfunctions whose eigenvalues lie in a unit band -- where the corresponding critical exponent is $\frac{2(d+1)}{d-1}$; the torus is better than a generic manifold precisely because lattice-point degeneracy thins its eigenspaces. In $d=2$ the conjecture with the $N^\epsilon$ loss is trivial, since $\#\mathcal{F}_{2,N}=r_2(N^2)\lesssim_\epsilon N^\epsilon$, where $r_2(m)$ counts representations of $m$ as a sum of two squares; so the substantive $d=2$ question is uniform boundedness. Cooke (1971), in the course of proving a two-dimensional Cantor-Lebesgue theorem (a result linking pointwise convergence of a trigonometric series to decay of its coefficients), and Zygmund (1974), answering a question of Fefferman, proved $M_{2,p}(N)\lesssim1$ for $2\le p\le4$ by a geometric argument on lattice points lying on a circle; no finite $p>4$ has been settled since, while at $p=\infty$ uniform boundedness fails because $M_{2,\infty}(N)\ge r_2(N^2)^{1/2}$ is unbounded. In $d=3$ only $p=4$ and $p=\infty$ are easy (both only with $N^\epsilon$ loss), and the critical exponent is $p_c=6$. The principal modern tools are: $\ell^2$ decoupling (Bourgain-Demeter 2015), a harmonic-analysis method that bounds a function with localized Fourier support by interpolating between an $L^2$ estimate and a maximal-type estimate on each piece of a decomposition of the support into small caps, and yields the conjecture with loss for $p\le\frac{2(d+1)}{d-1}$ in all dimensions; and the Hardy-Littlewood circle method -- which rewrites an average over integers as an integral over the unit circle, decomposed into arcs near rationals with small denominators -- applied to Weyl-type exponential sums (exponential sums with polynomial phase), which leads to Kloosterman and Salie sums -- complete exponential sums modulo $q$ obeying Weil's square-root-cancellation bound, which degenerates when $\gcd(\lambda,q)$ is large -- and yields the conjecture with loss for $p\ge\frac{2(d-1)}{d-3}$ when $d\ge4$ (the pure circle method gives $p\ge\frac{2d}{d-3}$). In the intermediate range around $p_c$, neither method suffices. Closely related objects: spectral projectors onto bands of width $\delta$ around frequency $N$ (equivalently quasimodes, functions that satisfy the eigenfunction equation only approximately, up to a small spectral error), whose $L^2\to L^p$ bounds are the subject of a companion conjecture of Germain and Rydin Myerson; and the additive energy $E_n(A)=\#\{(a_1,\dots,a_{2n})\in A^{2n}: a_1+\dots+a_n=a_{n+1}+\dots+a_{2n}\}$ of a lattice set $A$, which measures its additive structure and satisfies $E_n(\mathcal{F}_{d,N})=\|K\|_{L^{2n}}^{2n}$ for the projection kernel $K$, so additive-energy bounds are the combinatorial counterpart of $L^{2n}$ eigenfunction bounds.

Problem Statement

Determine the sharp growth in $N$ of $M_{d,p}(N)=\sup\|e_N\|_{L^p(\mathbb{T}^d)}/\|e_N\|_{L^2(\mathbb{T}^d)}$, the supremum over nonzero eigenfunctions $e_N$ of $\sqrt{-\Delta}$ with eigenvalue $2\pi N$ on the square torus $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$ (equivalently eigenfunctions of $-\Delta$ with eigenvalue $4\pi^2N^2$, $\lambda=N^2$; conventions parameterizing by $\lambda$ instead of $N$ differ by square roots, and 'uniformly bounded in $\lambda$' means uniformly bounded in $N$), for every dimension $d\ge2$ and every $2\le p\le\infty$. The common objective is to prove or disprove Bourgain's discrete restriction conjecture in full: $M_{d,p}(N)\lesssim_{\epsilon}N^{\epsilon}\Big(N^{\frac{d-2}{2}-\frac{d}{p}}+1\Big)\quad\text{for all } d\ge2, p\ge2, \epsilon>0,$ together with its sharp form: for each $(d,p)$, determine the infimum $\sigma(d,p)$ of exponents such that $M_{d,p}(N)\lesssim N^{\sigma(d,p)}$ -- conjecturally $\sigma(d,p)=\max(\frac{d-2}{2}-\frac{d}{p},0)$ for $d\ge5$ (no subpolynomial loss), and the same power with a possible subpolynomial $N^\epsilon$ loss for $d\le4$ -- and decide for which $(d,p)$ such a loss is necessary. As of September 2026 the open instances are: (a) $d=2$, uniform boundedness: for each finite $p>4$, is $M_{2,p}(N)$ uniformly bounded in $N$? (Known: yes for $2\le p\le4$; no at $p=\infty$.) (b) $d=3$: prove the conjectured bound for finite $p>4$, in particular at the critical exponent $p=6$, namely $M_{3,6}(N)\lesssim_\epsilon N^\epsilon$. (Known here: only $p=4$ and $p=\infty$, with loss, plus partial even-$p$ improvements.) (c) All $d\ge4$, critical and intermediate range: prove the conjectured bound for $\frac{2(d+1)}{d-1}<p<\frac{2(d-1)}{d-3}$, in particular at the critical exponent $p_c=\frac{2d}{d-2}$, where the conjecture reads $M_{d,p_c}(N)\lesssim_\epsilon N^\epsilon$; in $d=4$ this is the sharp $L^4$ bound, which would resolve the entire $d=4$ conjecture. (d) Sharpness of the loss: for $d\ge5$, remove the $N^\epsilon$ loss for $2<p\le\frac{2d}{d-4}$, finite $p$ (it is currently removed only for $p>\frac{2d}{d-4}$; below the critical exponent this is the uniform-boundedness statement $M_{d,p}(N)\lesssim1$); for $d\le4$, determine for which $(d,p)$ a subpolynomial loss is necessary.

References

1. R. Cooke (1971). A Cantor-Lebesgue theorem in two dimensions. Proceedings of the American Mathematical Society 30, 547-550. DOI: 10.1090/S0002-9939-1971-0282134-X. https://doi.org/10.1090/S0002-9939-1971-0282134-X
2. J. Bourgain (1993). Eigenfunction bounds for the Laplacian on the n-torus. International Mathematics Research Notices 1993(3), 61-66. DOI: 10.1155/S1073792893000066. https://doi.org/10.1155/S1073792893000066 (full text paywalled)
3. A. Zygmund (1974). On Fourier coefficients and transforms of functions of two variables. Studia Mathematica 50, 189-201. DOI: 10.4064/sm-50-2-189-201. https://doi.org/10.4064/sm-50-2-189-201
4. C. D. Sogge (1988). Concerning the $L^p$ norm of spectral clusters for second-order elliptic operators on compact manifolds. Journal of Functional Analysis 77, 123-138. DOI: 10.1016/0022-1236(88)90081-X. https://doi.org/10.1016/0022-1236(88)90081-X
5. J. Bourgain (1997). Analysis results and problems related to lattice points on surfaces. In: Contemporary Mathematics 208, 85-109. DOI: 10.1090/conm/208/02736. https://doi.org/10.1090/conm/208/02736 (survey recording the conjecture)
6. J. Bourgain (2013). Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces. Israel Journal of Mathematics 193, 441-458; preprint arXiv:1107.1129 (2011). DOI: 10.1007/s11856-012-0077-1. https://arxiv.org/abs/1107.1129 (Corollary 4 and the following Remark contain Bourgain's own restatement of the 1993 questions, including conjecture (3.8))
7. J. Bourgain and C. Demeter (2013). Improved estimates for the discrete Fourier restriction to the higher dimensional sphere. Illinois Journal of Mathematics 57(1), 213-227; preprint arXiv:1205.2414 (2012). DOI: 10.1215/ijm/1403534493. https://arxiv.org/abs/1205.2414 (states Conjecture 1, attributed to Bourgain 1993 and 1997, and the critical-index conjecture; proves the $p\ge\frac{2d}{d-3}$ range)
8. J. Bourgain and C. Demeter (2015). New bounds for the discrete Fourier restriction to the sphere in 4D and 5D. International Mathematics Research Notices 2015(11), 3150-3184; preprint arXiv:1310.5244 (2013). DOI: 10.1093/imrn/rnu036. https://arxiv.org/abs/1310.5244
9. J. Bourgain and C. Demeter (2015). The proof of the $\ell^2$ decoupling conjecture. Annals of Mathematics (2) 182(1), 351-389; preprint arXiv:1403.5335 (2014). DOI: 10.4007/annals.2015.182.1.9. https://arxiv.org/abs/1403.5335
10. P. Germain and S. L. Rydin Myerson (2022). Bounds for spectral projectors on tori. Forum of Mathematics, Sigma 10, paper e24. DOI: 10.1017/fms.2022.18. https://doi.org/10.1017/fms.2022.18
11. A. Mudgal (2022). Additive energies on spheres. Journal of the London Mathematical Society (2) 106(4), 2927-2958. DOI: 10.1112/jlms.12652. https://doi.org/10.1112/jlms.12652
12. P. Germain, S. L. Rydin Myerson and D. Pezzi (2025). Bounds for spectral projectors on the three-dimensional torus. Preprint arXiv:2508.05573. https://arxiv.org/abs/2508.05573 (states Conjecture B, Bourgain's three-dimensional eigenfunction conjecture, and surveys what is known in $d=3$)
13. D. Pezzi (2026). Sharp eigenfunction bounds on the torus for large $p$. Preprint arXiv:2603.10927 (v2, April 2026). https://arxiv.org/abs/2603.10927 (states the discrete restriction conjecture as Conjecture 1.1, attributed to Bourgain 1993; proves the first loss-free bounds for $d\ge5$)