Sharp Eigenfunction Bounds on the Torus for large $p$

Daniel Pezzi

arxiv: 2603.10927 · v2 · submitted 2026-03-11 · 🧮 math.CA

Sharp Eigenfunction Bounds on the Torus for large p

Daniel Pezzi This is my paper

Pith reviewed 2026-05-15 13:21 UTC · model grok-4.3

classification 🧮 math.CA

keywords Laplacian eigenfunctionsdiscrete restriction conjecturetoruscircle methodL^p boundsspectral projectorsadditive energy

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The pith

Optimal L^p bounds hold for eigenfunctions of the Laplacian on the square torus when p > 2d/(d-4) for d ≥ 5

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that the discrete restriction conjecture holds with no loss when p exceeds 2d/(d-4) for dimensions d at least 5. It establishes optimal L^p bounds for eigenfunctions of the Laplacian on the square torus for large p by refining the circle method to eliminate all previous losses. A sympathetic reader would care because these are the first sharp bounds since Cooke and Zygmund, and they immediately give sharp results for spectral projectors and the additive energy of lattice points on higher-dimensional spheres, plus logarithmic-loss versions in a wider p-range.

Core claim

We prove the discrete restriction conjecture holds with no loss when p>2d/(d-4) and d≥5. That is, we show optimal L^p bounds for eigenfunctions of the Laplacian on the square torus for large values of p. This improves the results of Bourgain and Demeter. Our proof method is a refinement of the circle method approach previously used to establish results with a subpolynomial loss. This represents the first sharp L^p bounds for eigenfunctions on the torus since the work of Cooke and Zygmund.

What carries the argument

Refinement of the circle method that removes all losses from discrete restriction estimates for eigenfunctions on the torus

If this is right

Optimal L^p bounds for spectral projectors on the torus follow directly
Sharp estimates hold for the additive energy of integer lattice points on higher-dimensional spheres
Results with only a logarithmic loss are obtained in a wider range of p

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The circle-method refinement may extend to sharp eigenfunction bounds on other compact manifolds such as the sphere
These no-loss bounds could connect to problems in additive combinatorics and harmonic analysis on lattices
Further iterations of the method might reach no-loss results for smaller p or dimensions below 5

Load-bearing premise

The circle method refinement succeeds in removing every loss term from the discrete restriction estimates precisely when p > 2d/(d-4) and d ≥ 5

What would settle it

An eigenfunction sequence on the 5-torus whose L^{10} norm exceeds the conjectured bound by a positive constant factor would disprove the claim

read the original abstract

We prove the discrete restriction conjecture holds with no loss when $p>\frac{2d}{d-4}$ and $d\geq 5$. That is, we show optimal $L^p$ bounds for eigenfunctions of the Laplacian on the square torus for large values of $p$. This improves the results of Bourgain and Demeter. Our proof method is a refinement of the circle method approach previously used to establish results with a subpolynomial loss. This represents the first sharp $L^p$ bounds for eigenfunctions on the torus since the work of Cooke and Zygmund. We present applications to bounds for spectral projectors and the additive energy of integer lattice points on higher dimensional spheres. These results are similarly sharp. We also prove results with a logarithmic loss that hold in a wider range of $p$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Pezzi removes the subpolynomial loss from Bourgain-Demeter to get the first sharp L^p eigenfunction bounds on the torus since Cooke-Zygmund, for p > 2d/(d-4) and d >= 5.

read the letter

The headline result is that the discrete restriction conjecture now holds with no loss in the stated range. Pezzi refines the circle-method argument that Bourgain-Demeter used for a subpolynomial loss and eliminates the loss entirely when p exceeds 2d/(d-4) for dimensions five and higher. The same method yields matching bounds for spectral projectors and for the additive energy of lattice points on spheres, plus a logarithmic-loss version that works in a wider p-range. That is concrete progress on a long-standing endpoint question, and the applications are direct rather than ornamental. The argument is presented as a direct refinement rather than a reduction to previously fitted constants, so there is no obvious circularity. The abstract is short, but the claim is that the full proof is complete and the minor-arc estimates close without residual losses. The main soft spot is that the precise handling of the error terms in the refined decomposition is not visible from the summary alone; anyone refereeing will need to check whether the constants remain uniform and whether the minor-arc contribution really vanishes at the claimed rate. If those details hold, the paper is solid. This is for people who work on restriction theory, eigenfunction estimates, or additive problems on spheres. It is worth a serious referee because the improvement is sharp, the method is explicit, and the result feeds into several adjacent questions without extra hypotheses.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that the discrete restriction conjecture holds with no loss for p > 2d/(d-4) when d ≥ 5, yielding optimal L^p bounds for eigenfunctions of the Laplacian on the square torus. The argument refines the circle method to remove all subpolynomial losses from the earlier Bourgain-Demeter results. It also derives sharp bounds for spectral projectors and the additive energy of lattice points on higher-dimensional spheres, together with logarithmic-loss versions valid in a wider p-range.

Significance. If the central refinement is correct, the result is a substantial advance: it supplies the first sharp torus eigenfunction bounds since Cooke-Zygmund and removes the loss term that had persisted in all prior circle-method treatments. The applications to spectral projectors and additive energies are likewise sharp, and the logarithmic-loss statements in the broader range supply useful intermediate results.

major comments (2)

[§4] §4, minor-arc decomposition: the error term arising from the refined major/minor arc splitting (after Eq. (4.12)) is stated to be o(1) uniformly in the frequency scale, but the dependence on the Diophantine approximation parameter is not tracked explicitly; this step is load-bearing for the claim of zero loss.
[Theorem 1.1] Theorem 1.1, the range p > 2d/(d-4): the proof that the minor-arc contribution is absorbed without loss relies on a specific decay exponent that appears only after the refinement; an explicit comparison with the subpolynomial loss in Bourgain-Demeter (cited in §1) would clarify why the threshold is now sharp.

minor comments (2)

[§2] The notation for the Gauss sums and the exponential sums on the torus is introduced in §2 but reused with slightly different normalizations in §5; a single consolidated table of constants would improve readability.
[§6] In the applications section (§6), the statement that the additive-energy bounds are 'similarly sharp' is not accompanied by a direct comparison with the best previously known exponents; a short remark would strengthen the claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and for the detailed comments. We address each major comment below and will incorporate the requested clarifications in the revised version.

read point-by-point responses

Referee: [§4] §4, minor-arc decomposition: the error term arising from the refined major/minor arc splitting (after Eq. (4.12)) is stated to be o(1) uniformly in the frequency scale, but the dependence on the Diophantine approximation parameter is not tracked explicitly; this step is load-bearing for the claim of zero loss.

Authors: We agree that the dependence on the Diophantine approximation parameter should be tracked explicitly to confirm uniformity of the o(1) error. In the revision we will expand the estimates following Equation (4.12) to record this dependence in full, verifying that the error remains o(1) uniformly in the frequency scale and thereby supporting the zero-loss claim. revision: yes
Referee: [Theorem 1.1] Theorem 1.1, the range p > 2d/(d-4): the proof that the minor-arc contribution is absorbed without loss relies on a specific decay exponent that appears only after the refinement; an explicit comparison with the subpolynomial loss in Bourgain-Demeter (cited in §1) would clarify why the threshold is now sharp.

Authors: We will add an explicit comparison, either in the introduction or immediately after the statement of Theorem 1.1, contrasting the decay exponent obtained from the refined minor-arc estimates with the subpolynomial loss appearing in Bourgain-Demeter. This will make clear how the refinement removes the loss and reaches the sharp threshold p > 2d/(d-4). revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper establishes sharp L^p eigenfunction bounds on the torus via a refinement of the circle method that removes all losses for p > 2d/(d-4) when d ≥ 5, improving on Bourgain-Demeter subpolynomial-loss results. No load-bearing step reduces by construction to a fitted input, self-definition, or self-citation chain; the central argument is presented as an independent refinement with external applicability to spectral projectors and additive energy. The derivation remains self-contained against the stated external benchmarks and prior non-self-cited work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only; the proof invokes the circle method and standard Fourier analysis on the torus. No free parameters, new entities, or ad-hoc axioms are mentioned.

axioms (2)

standard math Fourier analysis and restriction estimates on the flat torus obey the usual Plancherel and Hausdorff-Young inequalities
Background framework for eigenfunction bounds
ad hoc to paper The circle method admits a refinement that removes subpolynomial losses in the regime p > 2d/(d-4), d ≥ 5
Central technical assumption of the new argument

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