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Weyl formula improvement for product of Zoll manifolds

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Pith's one-line read On Cartesian products of Zoll manifolds, the Weyl counting remainder is improved from $O(\lambda^{d-1})$ to $O(\lambda^{d-1-(n-1)/(n+1)})$.

desk verdict Extends the Iosevich-Wyman polynomial Weyl improvement from products of spheres to products of arbitrary Zoll manifolds; the main idea is right and the lattice counting is careful, but two load-bearing steps are asserted rather than proved. read the letter →

arxiv 2506.02299 v1 pith:CS5R5HZY submitted 2025-06-02 math.AP

Yanfei Wang This is my paper
classification math.AP MSC 35P2058J5011P21
keywords WeyllawremainderZollmanifoldseigenvalueclustersweightedlatticepointsPoissonsummationMaslovindexproduct
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a polynomial improvement of the Weyl eigenvalue-counting law on any Cartesian product of Zoll manifolds. If $M=Z_{d_1}\times\cdots\times Z_{d_n}$ has total dimension $d=\sum d_i$, the remainder drops from the classical $O(\lambda^{d-1})$ to $O(\lambda^{d-1-(n-1)/(n+1)})$, a real polynomial gain for every $n\ge 2$. The reason it works is that each Zoll factor's spectrum splits into clusters whose multiplicities are a shifted monomial plus a lower-order error, so the product spectrum becomes a weighted lattice point count in a ball. The consequence is that the sharp remainder seen on every individual Zoll manifold is broken as soon as one takes products, in the same way it was already known to break for products of round spheres.

What carries the argument

A Zoll manifold is a compact Riemannian manifold all of whose geodesics are closed with a common period (normalized to $2\pi$). The machinery has two parts. First, spectral: a periodic first-order pseudodifferential operator $A=\sqrt{-\Delta+c}-T$ with spectrum $k+\alpha/4$ organizes eigenvalues into clusters, and the asymptotics of [6] plus the vanishing of the subprincipal symbol give the cluster multiplicity law $P_Z^d(t)=Ct^{d-1}+O(t^{d-3})$, $t=k+\alpha/4$. Second, arithmetic: the weighted lattice-point estimate of Proposition 3.3, proved by mollification, Poisson summation, and the Fourier bounds $|\widehat{\chi_B F}(\xi)|\lesssim\prod_i\langle\xi_i\rangle^{-2}$ and $|\widehat{\chi_B}(\xi)|\lesssim\langle\xi\rangle^{-(n+1)/2}$, optimized at $\varepsilon=\lambda^{-(n-1)/(n+1)}$, supplies the remainder term. The product theorem is the composition of these two ingredients.

What would settle it

Compute the cluster multiplicities of a Zoll manifold with explicit spectrum, for example complex projective space with its standard symmetric metric, and check whether they equal $C(k+\alpha/4)^{2n-1}+O(k^{2n-3})$ with no $k^{2n-2}$ term; a nonzero term at that order would falsify Lemma 1.3 and break the lattice reduction on which Theorem 1.4 rests.

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Extended reading notes

Core claim

Let $M=Z_{d_1}\times\cdots\times Z_{d_n}$ be a product of Zoll manifolds and let $N(\lambda)$ count the eigenvalues of $\sqrt{-\Delta}$ up to $\lambda$, with multiplicity. The central claim (Theorem 1.4) is $N(\lambda)=|B_d|(2\pi)^{-d}\operatorname{vol}(M)\lambda^d+O(\lambda^{d-1-(n-1)/(n+1)})$. The proof uses the cluster structure of each factor: the $k$-th cluster contains $P(t)=Ct^{d_i-1}+O(t^{d_i-3})$ eigenvalues with $t=k+\alpha_i/4$, and the product's counting function is therefore approximated by the weighted sum $\sum_{|m+y|\le\lambda}\prod_i C_i(m_i+\alpha_i/4)^{d_i-1}$ over an $n$-dimensional lattice. The remainder bound then follows from a weighted lattice-point estimate that is uniform in the shift vector $y$ made from the Maslov indices. This extends the known polynomial improvement for products of round spheres to the whole class of Zoll manifolds.

Load-bearing premise

The load-bearing premise is Lemma 1.3: on each Zoll factor the $k$-th cluster has exactly $C(k+\alpha/4)^{d-1}+O(k^{d-3})$ eigenvalues, which requires the first subleading coefficient in the cluster expansion to be exactly what formula (2.25) asserts; if that coefficient is wrong, the lattice sum picks up an error of size $\lambda^{d-1}$ and the polynomial improvement disappears.

Editorial extensions

If this is right

  • For any product of two or more Zoll factors, the $O(\lambda^{d-1})$ remainder is replaced by $O(\lambda^{d-1-(n-1)/(n+1)})$, a genuine polynomial improvement rather than just $o(\lambda^{d-1})$.
  • The exponent gain depends only on the number $n$ of factors, not on the dimensions, so adding more Zoll factors to a product gives a larger power saving.
  • Because one-dimensional Zoll manifolds are circles, the result also covers products of Zoll manifolds with flat tori, with the same exponent.
  • The Maslov shifts $\alpha_i/4$ enter the leading volume term and the constants but do not affect the remainder exponent, since the lattice estimate is uniform in the shift.
  • The sharp remainder on a single Zoll manifold is a one-factor phenomenon; the product structure is what makes the improvement appear.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof splits into a spectral ingredient (Lemma 1.3) and a lattice-counting ingredient (Proposition 3.3), so the same improvement should hold for any product of manifolds whose cluster multiplicities satisfy the same shifted polynomial law with no $t^{d-2}$ term, a generalization the paper leaves implicit.
  • The exponent $(n-1)/(n+1)$ is a uniform guarantee, not a sharp rate; for products containing tori, sharper lattice-point estimates in high dimensions suggest remainders as good as $O(\lambda^{d-2})$ are plausibly available.
  • A direct check of Lemma 1.3 on a Zoll manifold with explicit spectrum, such as complex projective space with its standard symmetric metric, would isolate the fragile step of the argument; if a $k^{d-2}$ term appeared in the cluster count, the whole reduction would need a new idea.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper claims that for a Cartesian product M = Z^{d_1} x ... x Z^{d_n} of n Zoll manifolds, the Weyl counting function satisfies N(lambda) = |B_{|d|}| (2pi)^{-|d|} vol(M) lambda^{|d|} + O(lambda^{|d|-1-(n-1)/(n+1)}). The strategy is to reduce the eigenvalue clusters of each Zoll factor, via Lemma 1.3, to shifted polynomial multiplicities C_i (m_i + alpha_i/4)^{d_i-1} + O(m_i^{d_i-3}), then to count weighted integer lattice points with weights prod m_i^{d_i-1}. The lattice-counting part is implemented through mollification, Poisson summation, and dyadic Fourier estimates, following the approach of Iosevich and Wyman for products of spheres. The main claim is Theorem 1.4, with Lemma 1.3 as the spectral input and Proposition 3.3 as the arithmetic input.

Significance. If the theorem is correct, it is a natural and valuable extension of the Iosevich-Wyman sphere result to all products of Zoll manifolds, and the improved exponent is explicit and uniform. The paper has several genuine strengths: the reduction of product clusters to a weighted lattice sum is conceptually clean; the proof of the ball-counting estimate in Proposition 3.3 is carefully executed with mollifiers, Poisson summation, and a dyadic decomposition; and the argument correctly identifies the Maslov shift as responsible for the cancellation that produces the improved remainder. The use of classical Duistermaat-Guillemin and Weinstein results is appropriate, and I do not see circularity. However, two load-bearing points are not established in the manuscript, and one of the stated estimates is false in a case covered by the theorem; these issues must be repaired before the paper can be accepted.

major comments (2)
  1. [Section 2, Eqs. (2.22)-(2.26)] The proof of Lemma 1.3 is incomplete at the one step on which the whole lattice reduction depends. Formula (2.25), asserting <c_1,1> = (1-n) integral_{sigma_A^{-1}(1)} sub(A - lambda_0 I), is introduced with 'we may prove ... as what we did in Theorem 2.4', but no derivation is supplied. Proposition 2.8 gives the existence of the expansion (2.21) but not the value of the second coefficient in the rho = 1 case, and Theorem 2.4 is stated for an unshifted operator Q rather than for the cluster problem with the phase shift lambda_0 and amplitude rho(bar B). The value of <c_1,1> is load-bearing: it provides exactly the cancellation that rewrites k^{n-1} + (n-1)lambda_0 k^{n-2} as (k + lambda_0)^{n-1} up to O(k^{n-3}). If the coefficient of k^{n-2} in (2.22) were anything else, formula (2.26) would contain a t^{n-2} term, and then in the product sum (3.9) one factor would contribute degree |d|-1 instead of |d|-2, leaving only the classical O(lambda^{|d|-1}) remainder and destroying Theorem 1.4. The sphere check validates the Maslov value in a special case but does not replace the missing general computation.
  2. [Section 3, Proposition 3.3 and Lemma 3.2] The annulus estimate (3.15), restated as (3.8) in Lemma 3.2, is not proved. After the statement of Proposition 3.3 the Poisson-summation argument treats only the ball sum I(lambda); the text never returns to J(lambda), and the proof of Lemma 3.2 stops after reducing to Proposition 3.3. This is not merely a missing detail, because (3.15) is false as stated. Take n=2, k=0, d=(1,1), and y=0; then F=1 and J(lambda) is the number of lattice points in the shell lambda <= |m| <= lambda + c/lambda. For lambda = sqrt(N) with N a sum of two squares having r_2(N) representations, all those representations lie in the shell, so J(lambda) >= r_2(N), which is unbounded as N grows. Thus Lemma 3.2(3.8) cannot hold in general. Since (3.8) is the estimate used to control the boundary term II in (3.5), the proof of Theorem 1.4 is unsupported unless the annulus statement is either proved under the hypotheses actually required or replaced by a different treatment of the torus factors; the false statement as written must be removed or restricted.
minor comments (5)
  1. [Throughout the introduction and Section 2] There are numerous typographical errors that should be corrected: 'piontwise', 'rusults', 'papaer', 'Lamma', 'comstructing', 'unitray', and 'psedo' are a sample.
  2. [Theorem 1.4] The formula contains '|B|d||', which should be '|B_{|d|}|', and the remainder exponent would be clearer as lambda^{|d|-1-(n-1)/(n+1)}.
  3. [Section 2, Proposition 2.2] In the proof, the symbol 'lambda9' appears twice where 'lambda_0' is intended.
  4. [Section 3, Eq. (3.4)] The vector delta_m = (c_1/m_1, ..., c_k/m_k, 0, ..., 0) is undefined when some m_i = 0; since only finitely many such indices occur, the definition should be made for m_i >= 1 and the finitely many exceptional terms handled separately.
  5. [Lemma 1.3] The phrase 'as a polynomial of k + alpha/4' suggests an exact polynomial identity, but the proof establishes an asymptotic expansion; the statement should say 'has the asymptotic form' to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central reduction uses external Duistermaat–Guillemin and Weinstein results, and the lattice-counting argument is self-contained.

full rationale

The paper's main theorem converts the product Weyl counting problem into a weighted lattice-point problem using Lemma 1.3, which asserts that the cluster multiplicities of a Zoll manifold have the form C t^{d-1} + O(t^{d-3}). This lemma is proved from the classical theorems of Duistermaat and Guillemin and of Weinstein, which are external to the paper and do not include the target product remainder. The specific subleading coefficient used in (2.25) is asserted rather than fully derived, but it is presented as a computation in the same external framework, not as a restatement of Theorem 1.4 and not as a fitted parameter. The lattice-counting portion in Section 3 is proved independently by mollification and Poisson summation; its error estimates do not depend on the Zoll constants beyond assuming the general polynomial shape from Lemma 1.3. The sphere case is used as a benchmark and the torus case is an external classical result. There are no fitted parameters renamed as predictions, no self-citation chain carrying the argument, and no target result assumed as an input. The unproved assertion around (2.25) is a mathematical rigor gap rather than a circularity, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No numbers are fitted to data or chosen to make the argument work: the constants C_i, c_i, and the Maslov shifts alpha_i/4 are geometric invariants fixed by the Zoll structure before the argument begins. No new physical or mathematical entities are postulated; the operators A, B_bar, T, U are standard microlocal constructions. The free-parameter count is zero. The axioms above are the external theorems and unproved local assertions on which the central claim rests.

assumptions (6)
  • domain assumption Weinstein's cluster asymptotic (Prop 2.8): ⟨psi_k, rho⟩ ~ (2pi)^{-n} sum_nu ⟨c_nu, rho⟩ k^{n-nu-1}, with the subleading coefficient given by (1-n) integrated over the cosphere of sub(A - lambda_0 I).
    Quoted from [21] with a sketch; all of Lemma 1.3's polynomial structure rests on it. Location: Section 2, around (2.21)-(2.25).
  • domain assumption Duistermaat-Guillemin spectral asymptotics (Theorem 2.4) with the density formula omega_1 = (1-n) times integral of sub(Q).
    Used to justify the singleton-at-zero singularity analysis and the subprincipal computation; cited from [6], not re-proved. Location: Section 2, Theorem 2.4.
  • standard math sub(Delta) = 0 and sub(A) = 0 (Lemma 2.5 and Corollary 2.6).
    Standard fact in microlocal analysis; the paper gives a normal-coordinate argument. Needed for the Maslov-shift computation in (2.25)-(2.26).
  • domain assumption No fixed points of the geodesic flow on (0, 2pi): (2.19) f_t(x) != f_tau(x) for all t, tau in (0, 2pi).
    Asserted as 'we can make the following assumptions from now'; needed so that Tr(S(t)) has a single singularity at t = 0. Not proven to follow from the Zoll definition, since some geodesics could close after a proper divisor of the common period.
  • ad hoc to paper Zoll cluster multiplicity polynomial (Lemma 1.3): P_Z(k + alpha/4) = C (k + alpha/4)^{n-1} + O(k^{n-3}).
    Load-bearing: the entire lattice reduction (3.9) uses this exact shifted polynomial form. The proof reduces it to the quoted DG/Weinstein results, with the subleading coefficient asserted at (2.25). If false, Theorem 1.4 fails.
  • ad hoc to paper Weighted annulus estimate (Prop 3.3 (3.15) and Lemma 3.2 (3.8)): J(lambda) = O(lambda^{|d|-2}).
    Stated in the proposition but no proof is given in the text; it is used to bound the boundary piece II in Theorem 1.4. A weaker bound follows from (3.14), so this may be patchable, but as written it is an unproved input.

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Pith. "Pith review of Weyl formula improvement for product of Zoll manifolds." pith.science (2026). https://pith.science/paper/CS5R5HZY

@misc{pith2026250602299,
  author       = {Pith},
  title        = {Pith review of: Weyl formula improvement for product of Zoll manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CS5R5HZY}},
  note         = {Machine review of arXiv:2506.02299}
}
abstract

Iosevich and Wyman have proved in ~\cite{IoWy} that the remainder term in classical Weyl law can be improved from $O(\lambda^{d-1})$ to $o(\lambda^{d-1})$ in the case of product manifold by using a famous result of Duistermaat and Guillemin. They also showed that we could have polynomial improvement in the special case of Cartesian product of round spheres by reducing the problem to the study of the distribution of weighted integer lattice points. In this paper, we show that we can extend this result to the case of Cartesian product of Zoll manifolds by investigating the eigenvalue clusters of Zoll manifold and reducing the problem to the study of the distribution of weighted integer lattice points too.

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