{"id":"931c5ceb-137a-407b-b8d0-3b955256ffe8","arxiv_id":"2412.14059","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors improve Weyl remainder estimates for balls and spherical shells in all dimensions and prove new facts about zeros of Bessel cross-products.","lead":"This paper sharpens the error term in the asymptotic eigenvalue count for the Dirichlet and Neumann Laplacian on balls and spherical shells in all dimensions, improving the best known exponent from about 0.6298 to 0.628966 when a rationality condition holds. It also proves new uniform asymptotics and real-simplicity results for zeros of Bessel-function cross-products, which are useful in spectral geometry and analytic number theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The improved 2θ* bounds hinge on Theorem 6.1, whose nondegeneracy-free extension is only sketched and deferred to the unpublished [27]; that missing verification is the load-bearing risk.","rationale":"The reader identified the same load-bearing point, and I agree. I checked the applications of Theorem 6.1: for the dyadic sums in Lemma 5.1, T=M^{5/3}μ^{1/3} and M=2^j μ^{2θ*}; the range condition (6.2) is satisfied for each dyadic block, and the derivative-size assumptions on the functions F in (5.9) and (5.16) appear consistent with the cusp behavior of H and T. Thus the internal lattice-point reduction is not where the argument is weakest. The genuinely load-bearing assumption is the claimed extension of [27, Theorem 4.2] that removes the nondegeneracy condition (6.1). The paper itself labels that proof sketchy and defers details to an unpublished preprint, and no machine-checked or independently verified proof is supplied. I would therefore keep the reader's CONDITIONAL verdict: the stated 2θ* bounds follow if the extension is valid; otherwise only the O(μ^{d−2+2/3}) shell bound and the Huxley-based estimates remain secure.","tokens_in":41174,"tokens_out":17084,"duration_ms":151394,"concrete_test":"Require a complete write-up of the proof of Theorem 6.1 that does not cite [27, Theorem 4.2] as a black box. Concretely, trace every use of Huxley's Lemmas 3.3 and 3.4 in the second-spacing step and exhibit the parameter ranges, showing that a positive lower bound on F'F'''−3F''^2 is never invoked when M∈[T^{141/328+c}, T^{1/2}]. If any step needs that lower bound, Theorem 6.1 fails and the 2θ* claims are unsupported. A complementary indicative check is to test (6.3) numerically at moderate T for a model F whose ratio u=F''/F' has u'(x0)=0, e.g. F'=exp(1+(x−1)^3), and compare against T^{θ*+ε}; a clear violation would be decisive, though agreement is not proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 6.1 is the sole conduit for the improved exponent. Both the ball bound (1) and the rational-shell part of (2) in Theorem 1.1 are obtained by applying (6.3) to the rounding-error sums in Lemma 5.1 and Theorem 5.5. What distinguishes those applications from Huxley's older bound is the assertion, in the last paragraph of the proof in Section 6, that condition (6.1) is unnecessary for M in (6.2). The proof is explicitly a sketch: the first-spacing estimate is quoted from [27, Prop. 3.1], the second-spacing estimate is quoted from Huxley's Lemmas 3.3 and 3.4, and the sentence that (6.1) is only needed in the case M ≳ T^{181/328} is asserted without a displayed derivation. Since 181/328 ≈ 0.5518 > 1/2, that subcase is trivially disjoint from (6.2), but the real question is whether (6.1) is also used in other, non-displayed subcases of the second-spacing argument. The manuscript does not reproduce [27, Theorem 4.2] or its proof, and [27] is an unreviewed preprint with no machine-checked formalization. If the nondegeneracy-free extension fails, the ball bound reverts to the Huxley-based 131/208 exponent and the rational-shell improvement disappears; only the general shell bound O(μ^{d−2+2/3}) remains. This is a provenance/correctness risk rather than an internal inconsistency: the reduction from eigenvalues to weighted lattice sums is detailed and structurally coherent.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops uniform asymptotics and zero-counting results for cross-products of Bessel functions and ultraspherical Bessel functions, and applies them to the two-term Weyl law for Euclidean balls and spherical shells in R^d. Theorem 1.1 claims a remainder O_epsilon(mu^{d-2+2theta_*+epsilon}) for balls in all dimensions and both Dirichlet and Neumann boundary conditions, with 2theta_* approximately 0.628966, improving the previous 131/208 approximately 0.629807; for shells it gives O(mu^{d-2+2/3}) in general and the same improved exponent when pi^{-1} arccos(r/R) is rational. The proof reduces eigenvalue counting to weighted planar lattice-point counting via the Bessel zero asymptotics, and then applies a new exponential-sum estimate (Theorem 6.1).","tokens_in":41431,"tokens_out":9614,"duration_ms":78598,"significance":"If the proof of Theorem 6.1 is completed, the paper's results are significant: they improve the best known upper bounds for the Weyl remainder on balls in all dimensions, extend the planar annulus results to higher-dimensional shells in both Dirichlet and Neumann cases, and provide a new family of Bessel-function estimates of independent interest. The long Bessel-function part (Sections 2-4) contains detailed, apparently sound proofs of uniform zero asymptotics (Theorems 2.20 and 4.13) and of real-simplicity of zeros (Theorem 3.3). The main weakness is the reliance of the improved exponent on Theorem 6.1, whose proof is only sketched and whose decisive input is the unpublished preprint [27] by one of the authors; this is a correctness-risk concern rather than an internal inconsistency.","major_comments":[{"comment":"The improved exponent in Theorem 1.1 depends entirely on Theorem 6.1, but its proof is explicitly a sketch. The decisive assertion that condition (6.1) can be dropped for M in the range (6.2) is made in the final paragraph of the proof: it is stated that (6.1) is only needed in the case M ≳ T^{181/328} and that this is disjoint from (6.2), but the subcase analysis of Huxley's Lemmas 3.3 and 3.4 that would demonstrate that (6.1) is not used elsewhere is not displayed. Since [27] is an unreviewed preprint by one of the authors and its Theorem 4.2 is not reproduced, the reader cannot verify the key estimate (6.3). This is a load-bearing gap: if the nondegeneracy-free extension fails, the ball bound reverts to the Huxley exponent and the rational-shell improvement disappears. The authors should either give a complete proof of Theorem 6.1 or state the improved bounds as conditional on [27, Theorem 4.2].","section":"Section 6, Theorem 6.1"},{"comment":"The application of Theorem 6.1 requires the functions F defined in (5.9) and (5.16) to satisfy |F^{(j)}(x)| ≍ 1 for j=1,2,3. For (5.16) the verification is deferred to \"the size of derivatives of T (in [14, Lemma 4.7])\", but the third-derivative bound is not shown, and the constant in the lower bound matters for the dyadic summation. Since this condition is part of the hypothesis of Theorem 6.1, the proof of Lemma 5.1 should explicitly verify it for both (5.9) and (5.16).","section":"Lemma 5.1, equations (5.9) and (5.16)"}],"minor_comments":[{"comment":"The sentence \"Even thought the planar domain for the corresponding lattice point problem could be bad\" contains a typo: \"thought\" should be \"though\".","section":"Introduction, page 3"},{"comment":"The leading coefficient is typeset as \"2d(Γ(d/2 + 1))2\"; this should be 2^d (Γ(d/2 + 1))^2 to avoid ambiguity.","section":"Theorem 5.4 and its proof"},{"comment":"Reference [27] is an arXiv preprint; it should be marked as such, with the version number and date, rather than appearing in the same format as journal articles.","section":"References"},{"comment":"The summation ∑_{m=M}^{M_2} requires M to be an integer; either state that M is a positive integer or replace the lower limit by ⌊M⌋.","section":"Theorem 6.1"},{"comment":"The discussion of Huxley's interpretation of [14] is a scholarly dispute rather than a mathematical result; consider moving it to a separate note or footnote so that the main text remains focused on the mathematical content.","section":"Remark 5.6"}],"recommendation":"major_revision","confidential_remarks":"The main concern is provenance: [27] is an unreviewed arXiv preprint coauthored by the last author, and the present paper's key estimate is a nondegeneracy-free extension of a theorem in that preprint. The authors' own proof of Theorem 6.1 is labeled \"sketchy\" and defers details to [27]. I would advise the editor to require either a full proof of Theorem 6.1 in this paper or an updated, publicly refereed version of [27] before accepting the improved exponent as unconditional. Also, Remark 5.6 is a pointed criticism of a published paper by Huxley; the authors should be asked to double-check the accuracy of their reading of [20] and to phrase any clarification in a less polemical style."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a serious extension of the authors' earlier work on Weyl remainders for balls and annuli, now covering spherical shells in all dimensions and both Dirichlet and Neumann boundary conditions, and it improves the best known remainder exponent from 131/208 ≈ 0.6298 to 2θ* ≈ 0.628966. The reduction from eigenvalue counting to weighted planar lattice point counting is coherent, and the Bessel-function analysis (uniform asymptotics for zeros of cross-products of derivatives of ultraspherical Bessel functions, and the real-simplicity result in Theorem 3.3) is detailed and mostly self-contained.\n\nWhat is new: the shell case with Neumann conditions is not in the prior annulus/ball papers, and carrying the Dirichlet program over to all dimensions with the multiplicity weights is a genuine extension, not a routine rerun. The proof has no circularity: the exponent θ* solves a fixed equation, not fitted to eigenvalues, and the lattice-point reduction is independent of the target bound.\n\nThe soft spot is Theorem 6.1. The improved ball bound and the rational-slope shell bound both pass through this estimate, and the proof is explicitly a sketch. It defers the first-spacing bound to [27, Prop. 3.1], a preprint by the last author, and the claim that the nondegeneracy condition (6.1) is unnecessary in the range M ∈ [T^{141/328+c}, T^{1/2}] is asserted without a displayed derivation. I believe the conclusion may well be true, and the heuristic that (6.1) is only needed for M ≳ T^{181/328} is plausible, but the burden is real. Since [27] is not machine-checked and not reproduced, a referee should demand a complete proof of Theorem 6.1 or a detailed derivation of (6.3) before the improved exponent is considered established. If Theorem 6.1 fails, the paper still proves the general shell bound O(μ^{d-2+2/3}) and the Huxley-based results, so it is not empty.\n\nMinor: Remark 5.6 biting Huxley is tangential; it does not affect the math but is a bit of a distraction.\n\nWho this is for: spectral asymptotics people and anyone interested in exponential sums and lattice point counting. Deserves a serious referee; I would send it out, with the request that the referee specifically check Section 6 and the dependence on [27].","headline":"Strong technical paper whose improved Weyl exponent rests on a sketched estimate outsourced to an unpublished preprint by the last author; referee before trusting the new bound.","tokens_in":42062,"tokens_out":3568,"would_cite":true,"duration_ms":31598,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P20","42B20","11P21","33C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes that the Weyl remainder for balls and spherical shells in every dimension $d \\ge 2$ is bounded by $\\mu^{d-2+2\\theta_*+\\varepsilon}$, with $2\\theta_* \\approx 0.628966$, in both Dirichlet and Neumann cases.","keywords":["Weyl law","eigenvalue counting","Bessel functions","ultraspherical Bessel functions","spherical shells","Gauss circle problem","weighted lattice point counting","decoupling estimates"],"falsifier":"Take $F(x)=\\sqrt{x+1}$ on $[1/2,2]$; it satisfies $|F^{(j)}(x)| \\asymp 1$ for $j=1,2,3$ while $F'F'''-3(F'')^2 \\equiv 0$, so the nondegeneracy condition of the cited preprint fails exactly in the way Theorem 6.1 claims is harmless. For $M = T^{141/328+c}$ with a small $c>0$, evaluate the rounding-error sum in (6.3) for large dyadic $T$; if any value exceeds $T^{\\theta_*+\\varepsilon}$, Theorem 6.1 is false and the improved Weyl bounds do not follow.","tokens_in":40904,"feed_emoji":"📐","tokens_out":9325,"duration_ms":78780,"temperature":0.7,"pith_summary":"This paper proves new upper bounds for the remainder in Weyl's law for the Dirichlet and Neumann Laplacians on Euclidean balls and spherical shells in all dimensions $d \\ge 2$. For balls the remainder is shown to be $O_\\varepsilon(\\mu^{d-2+2\\theta_*+\\varepsilon})$, and for spherical shells it is $O(\\mu^{d-2+2/3})$ in general, improving to $O_\\varepsilon(\\mu^{d-2+2\\theta_*+\\varepsilon})$ when $\\pi^{-1}\\arccos(r/R)$ is rational. The exponent $2\\theta_* = 0.628966\\ldots$ improves the older $131/208 = 0.629807\\ldots$. To reach these bounds, the paper develops uniform asymptotics and zero-counting results for cross-products of Bessel functions and of derivatives of ultraspherical Bessel functions, results that are presented as independently useful. If the central technical estimate holds, the shell results are new in all dimensions and the ball results cover the Neumann case for the first time.","feed_headline":"Weyl-law remainders for balls and shells reach exponent 0.628966","feed_subtitle":"In every dimension, the new bound beats the old 131/208 for Dirichlet and Neumann spectra alike.","key_machinery":"The load-bearing object is the planar domain $\\Omega$ bounded by the graph of $G(x)=Rg(x/R)-rg(x/r)$ and its Minkowski functional $F$; the positive zeros of the Bessel cross-products $f_\\nu$, $g_\\nu$, and $h_{\\nu,\\delta}$ are approximated as $F(\\nu,k+\\tau_{\\nu,k})$ with uniform remainder terms. That approximation turns eigenvalue counting into weighted lattice-point counting in $\\mu\\Omega$, and the exponent $2\\theta_*$ enters through Theorem 6.1, an estimate for sums of rounding errors with the sawtooth function that extends the recent Gauss-circle bound to the functions arising in this spectral problem.","core_discovery":"The central claim is Theorem 1.1: for $d \\ge 2$ and any $\\varepsilon > 0$, the Weyl remainder for a ball satisfies $R_B(\\mu) = O_\\varepsilon(\\mu^{d-2+2\\theta_*+\\varepsilon})$, and for a spherical shell satisfies $R_S(\\mu) = O(\\mu^{d-2+2/3})$, improved to $O_\\varepsilon(\\mu^{d-2+2\\theta_*+\\varepsilon})$ when $\\pi^{-1}\\arccos(r/R) \\in \\mathbb{Q}$, in both Dirichlet and Neumann cases. The proof converts eigenvalue counting into weighted planar lattice-point counting in a domain built from the function $G(x)=Rg(x/R)-rg(x/r)$, with $g(x)=(\\sqrt{1-x^2}-x\\arccos x)/\\pi$, and then applies a recent exponential-sum estimate from the Gauss circle problem, extended in Theorem 6.1 to the functions that arise here. The paper also establishes that all zeros of the relevant cross-product of derivatives of ultraspherical Bessel functions are real and simple, and counts those zeros inside a large circle.","pith_inferences":["Editorial inference: The same weighted lattice-point reduction should carry the $2\\theta_*$ exponent to Robin boundary conditions on balls and spherical shells, since the boundary-condition parameter enters only through the Bessel cross-product whose zeros are already treated with uniform estimates.","Editorial inference: The uniform zero approximations developed here could be used to certify numerical eigenvalue computations for spherical shells with explicit error bounds, a by-product the paper does not pursue.","Editorial inference: If Theorem 6.1 is fully verified, any future improvement of the Gauss circle or Dirichlet divisor exponent would automatically improve these Weyl remainders through the same reduction, making the spectral problem a direct consumer of exponential-sum progress.","Editorial inference: The reverse inequality noted in Remark 4.3 suggests that sharper Weyl bounds on shells could in turn yield new lattice-point estimates for the planar domain $\\Omega$, reversing the usual direction of transfer."],"forward_implications":["For every $d \\ge 2$, the Dirichlet and Neumann eigenvalue-counting remainders for balls improve to $O_\\varepsilon(\\mu^{d-2+2\\theta_*+\\varepsilon})$, beating the previously known $131/208$-based bound in all dimensions.","For spherical shells, the general remainder $O(\\mu^{d-2+2/3})$ holds for both boundary conditions, and under the rationality condition $\\pi^{-1}\\arccos(r/R) \\in \\mathbb{Q}$ it improves to $O_\\varepsilon(\\mu^{d-2+2\\theta_*+\\varepsilon})$, a result that is new in dimensions $d \\ge 3$.","The uniform zero approximations, such as $x''_{\\nu,k} = F(\\nu,k+\\tilde\\tau_{\\nu,k}) + O((\\nu+k)^{-1})$ in appropriate ranges, provide explicit approximations of eigenvalues with error terms independent of the quantum numbers $\\nu$ and $k$.","The proof that the cross-product of derivatives of ultraspherical Bessel functions has only real and simple zeros, with an exact count inside large circles, supplies a spectral ingredient that can be reused in other boundary-value problems with spherical symmetry.","The reduction to weighted planar lattice-point counting, with weights coming from eigenvalue multiplicities, gives a template for transferring further improvements of the Gauss circle problem exponent to Weyl remainders."],"supporting_citations":[{"why":"Supplies the $\\theta_*$ exponential-sum estimate whose restricted-range extension, Theorem 6.1, carries the improved Weyl exponent.","marker":"[27]"},{"why":"Provides the predecessor $131/208$ lattice-point bound and the second-spacing lemmas used in the proof of Theorem 6.1 and of the $O(\\mu^{d-2+2/3})$ results.","marker":"[19]"},{"why":"Establishes the planar annulus Weyl formula and the reduction from eigenvalue counting to weighted planar lattice counting that the shell argument generalizes.","marker":"[14]"},{"why":"Establishes the ball Dirichlet Weyl bound and the weighted lattice-point machinery reused for balls in all dimensions.","marker":"[13]"},{"why":"Supplies uniform enclosures for the phase and zeros of Bessel functions, used to prove that the relevant cross-product zeros are real and simple for small order.","marker":"[12]"},{"why":"Provides the classical theorem that cross-product Bessel functions have only real simple zeros, together with the zero count inside a large circle used in Section 3.","marker":"[5]"}],"fun_headline_variants":["Improved Weyl remainders for balls and shells in all dimensions","New Weyl-law exponent for balls and shells: 0.628966","Balls and shells: Weyl remainder bound beats 131/208","Bessel zeros sharpen Weyl remainders on balls and shells","Gauss circle trick yields better Weyl remainders for balls"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The improved exponent $2\\theta_*$ stands on Theorem 6.1, a sketched extension of an exponential-sum estimate from the preprint [27] that removes a nondegeneracy condition in the range $T^{141/328+c} \\le M \\le T^{1/2}$; if that extension fails, only the weaker $O(\\mu^{d-2+2/3})$ shell bound and the older $131/208$-type bounds remain secure.","fun_headline_variants_meta":{"raw":{"variants":["Improved Weyl remainders for balls and shells in all dimensions","New Weyl-law exponent for balls and shells: 0.628966","Balls and shells: Weyl remainder bound beats 131/208","Bessel zeros sharpen Weyl remainders on balls and shells","Gauss circle trick yields better Weyl remainders for balls"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00136,"raw_usage":{"total_tokens":5536,"prompt_tokens":979,"completion_tokens":4557,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":595,"completion_tokens_details":{"reasoning_tokens":4466}},"tokens_in":595,"tokens_out":4557,"duration_ms":35919,"temperature":1.0,"reasoning_tokens":4466,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:31:14.632447+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $F(x)=\\sqrt{x+1}$ on $[1/2,2]$; it satisfies $|F^{(j)}(x)| \\asymp 1$ for $j=1,2,3$ while $F'F'''-3(F'')^2 \\equiv 0$, so the nondegeneracy condition of the cited preprint fails exactly in the way Theorem 6.1 claims is harmless. For $M = T^{141/328+c}$ with a small $c>0$, evaluate the rounding-error sum in (6.3) for large dyadic $T$; if any value exceeds $T^{\\theta_*+\\varepsilon}$, Theorem 6.1 is false and the improved Weyl bounds do not follow.","supporting_citations":[{"cited_title":"and Wang, Z., The Weyl formula for planar annuli , J","cited_arxiv_id":null,"evidence_quote":"Establishes the planar annulus Weyl formula and the reduction from eigenvalue counting to weighted planar lattice counting that the shell argument generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the ball Dirichlet Weyl bound and the weighted lattice-point machinery reused for balls in all dimensions."},{"cited_title":"and Sher, D., Uniform enclosures for the phase and zeros of Bessel functions and their derivative , SIAM J","cited_arxiv_id":null,"evidence_quote":"Supplies uniform enclosures for the phase and zeros of Bessel functions, used to prove that the relevant cross-product zeros are real and simple for small order."},{"cited_title":"A., Remarks on the zeros of cross-product Bessel functions, J","cited_arxiv_id":null,"evidence_quote":"Provides the classical theorem that cross-product Bessel functions have only real simple zeros, together with the zero count inside a large circle used in Section 3."}],"review_version":1}