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Asymptotic evaluations of generalized Bessel function of order zero related to the p-circle lattice point problem

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Pith's one-line read The paper proves uniform decay $J_0^p(\eta)=O(|\eta|_p^{-1/2})$ on compact sectors for $0<p<1$ and $p=2$, and the whole-plane uniform rate $O(|\eta|_p^{-p/2})$ when $2/p\in\mathbb{N}\setminus\{1,2\}$.

desk verdict Genuinely new uniform asymptotics for a generalized Bessel function; the core proof is sound but the paper overstates the role of its densest proposition. read the letter →

arxiv 2411.10850 v4 pith:7XN3ZM5D submitted 2024-11-16 math.NT math.CA

Masaya Kitajima This is my paper · ORCID
classification math.NTmath.CA MSC 11P2133C1042B20
keywords latticepointproblemp-circlesuperellipsegeneralizedBesselfunctionuniformasymptoticestimatesoscillatoryintegralvanderCorputlemma
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 is about the order-zero generalized Bessel function $J_0^p$ that enters harmonic-analytic formulas for counting lattice points inside the $p$-circle $\{x: |x_1|^p + |x_2|^p = r^p\}$. The author proves that on angular sectors bounded away from the coordinate axes, $J_0^p(\eta)=O(|\eta|_p^{-1/2})$ uniformly for every $0

What carries the argument

The central object is the oscillatory-integral representation of Proposition 2.1, which rewrites $J_0^p(\eta)$ as a constant times the sum of four integrals $\int_0^{\pi/2}e^{i\lambda(\pm f_{p,\varphi})}\psi^{[p]}(\theta)d\theta$ and $\int_0^{\pi/2}e^{i\lambda(\pm g_{p,\varphi})}\psi^{[p]}(\theta)d\theta$, with weights $\psi^{[p]}(\theta)=(\cos\theta\sin\theta)^{2/p-1}$. Near an axis the relevant phases take the form $F_{p,\delta}(\theta)=\delta\cos^{2/p}\theta+\sin^{2/p}\theta$ and $G_{p,\delta}(\theta)=\delta\sin^{2/p}\theta-\cos^{2/p}\theta$. The proof mechanism is real-variable stationary phase: away from axes a single interior stationary point with nonzero second derivative gives the $|\eta|_p^{-1/2}$ rate, while on the axes the endpoint stationary points have nonvanishing $2/p$-th derivative exactly when $2/p$ is an integer; Proposition 2.8 extends this nonvanishing control to the $\delta$-dependent stationary point, so van der Corput's lemma forces the uniform $|\eta|_p^{-p/2}$ decay.

What would settle it

For a single admissible case, say $p=2/3$ (so $2/p=3$), test the claimed $O,\Omega(1)$ bound by expanding $F_{p,\delta}(\pi/2-\theta_\delta)$; the leading constant in $F_{p,\delta}^{(3)}$ must be nonzero, and if it vanishes the uniform whole-plane bound collapses. A direct numerical check is to compute $|\eta|_p^{p/2}|J_0^p(\delta\lambda,\lambda)|$ for large $\lambda$ and $\delta\to0^+$; unboundedness along this wedge would falsify Theorem 1.5.

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Extended reading notes

Core claim

On the author's own terms, the discovery is that the asymptotic decay of $J_0^p$ is governed by the $p$-radius $|\eta|_p$ and by the exponent $p/2$ near the axes: for $0<p<1$ or $p=2$, the decay away from the axes is uniformly $O(|\eta|_p^{-1/2})$, while for $2/p\in\mathbb{N}\setminus\{1,2\}$ the same order of uniformity holds on all of $\mathbb{R}^2$ with the slower algebraic rate $O(|\eta|_p^{-p/2})$. The delicate point is that as the direction approaches an axis the stationary point of the phase slides toward an endpoint; Proposition 2.8 controls the $2/p$-th derivative of the phase at that moving point uniformly in $\delta$, and that control is what lets van der Corput's lemma give a uniform bound right up to the axes. For $p=2$ the statement reduces to the familiar $J_0(|x|)=O(|x|^{-1/2})$.

Load-bearing premise

The whole-plane bound depends on Proposition 2.8, which asserts that the $2/p$-th derivative of the phase $F_{p,\delta}$ at the $\delta$-dependent stationary point stays bounded between two positive constants as the direction approaches an axis; if that lower bound failed, the uniform $|\eta|_p^{-p/2}$ estimate on all of $\mathbb{R}^2$ would not follow from the proof given.

Editorial extensions

If this is right

  • For each $p=2/N$ with integer $N\ge3$, the estimate $J_0^p(\eta)=O(|\eta|_p^{-p/2})$ holds uniformly over all directions, so the axis directions no longer form an exceptional set for these $p$.
  • Together with Theorem 1.4, the result gives the whole-plane analogue of the angular-sector rate $|\eta|_p^{-1/2}$ exactly for the family $2/p\in\mathbb{N}$, with the necessarily slower exponent $p/2$ near the axes.
  • In Section 3 the bound for order zero, with $q_0^p=p/2$, is the first input in the criterion for absolute convergence of the generalized series; supplying the corresponding bounds for positive orders would identify the admissible range of $\beta$ in Theorem 3.1.
  • If the conjectured oscillatory-integral representation for positive integer orders is proved, the same van der Corput argument immediately yields uniform $O(|\eta|_p^{-q_n^p})$ bounds for $J_n^p$.
  • The compact-set theorem stops short of $p=1$ (Remark 2.6), so the method does not cover the diamond-shaped $\ell^1$-ball case.

Reading between the lines

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

  • Because whole-plane uniformity is achieved precisely when $2/p$ is an integer, the axis obstruction appears tied to the fractional smoothness of $\cos^{2/p}\theta$ and $\sin^{2/p}\theta$ at the endpoints; for non-integral $2/p$, directional dependence of the decay near the axes would be expected and could be probed numerically.
  • For the family $p=2/N$, the lattice-point error of the corresponding superellipse should become accessible through the series in Theorem 1.3 once the positive-order analogues are in hand, and the paper's closing remarks suggest that only orders $\omega=1,2$ may be needed.
  • Proposition 2.8 is numerically checkable: for a small admissible $p$, the leading constant in the $O,\Omega(1)$ bound for $F_{p,\delta}^{(2/p)}(\pi/2-\theta_\delta)$ should be nonzero, and its vanishing would destroy the uniform whole-plane theorem.
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Editorial analysis

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

Referee Report

0 major / 6 minor

Summary. The paper studies the generalized Bessel function J_0^p associated with the p-circle lattice point problem. The main results are asymptotic estimates as |η|_p → ∞: Theorem 1.4 gives uniform O(|η|_p^{-1/2}) on compact subsets of the open quadrants for 0<p<1 or p=2, and Theorem 1.5 gives uniform O(|η|_p^{-p/2}) on all of R^2 for p with 2/p ∈ N\{1,2}, together with the classical O(|η|^{-1/2}) for p=2. The proofs use an oscillatory integral representation (Proposition 2.1), stationary-phase expansions (Lemma 2.5), van der Corput's lemma (Lemma 2.7), and a lengthy derivative estimate (Proposition 2.8). The paper also sketches consequences for the lattice point program in Section 3.

Significance. If correct, Theorem 1.5 provides the first uniform-on-R^2 decay estimates for this generalized Bessel function in the p<1 regime, a relevant step in the author's harmonic-analytic approach to the p-circle lattice point problem. The proofs are self-contained and rely on standard oscillatory-integral tools; there are no fitted parameters and the desired estimates are not used as input. The main claims appear correct, and the stress-test concern about Proposition 2.8 does not actually threaten Theorem 1.5, because the uniform lower bound in (2.8) alone suffices after re-selecting δ'. The paper is honest about limitations (p=1 is excluded, and positive-order J_ω^p is not treated). The exposition is workmanlike, though several statement-level inconsistencies and typos need correction.

minor comments (6)
  1. [Abstract / Theorem 1.4] The abstract states that the compact-set estimates hold for 0<p≤1, but Theorem 1.4 and Remark 2.6 explicitly exclude p=1, which is a genuine exceptional case (for p=1 the phase can be constant on the diagonal). Please change the abstract to 0<p<1.
  2. [Theorem 1.5] The hypothesis '2/p are the natural numbers other than 2' is ambiguous: it can be read as 2/p ∈ N\{2}, which includes p=2, but the display then treats p=2 separately. Please rephrase, for example 'For p=2 and for 0<p<1 with 2/p ∈ N\{1,2}, ...'.
  3. [Section 2.2, proof of Theorem 1.5] The proof invokes Proposition 2.8, but the uniform nonzero bound in (2.8) on [0,a] already suffices after re-selecting δ' so that the δ-dependent stationary point π/2−θδ lies in [0,a]. Consider simplifying the proof or clarifying what Proposition 2.8 adds.
  4. [Section 2.3, equation (2.15)] The condition in (2.15) is written as 'if 1 ≤ k, that is, 0 ≤ (1−kp)/(1−p) ≤ 1'; the intended range appears to be 1 ≤ k ≤ 1/p, since for larger k the displayed expression O(δ^{(1−kp)/(1−p)}) would be a growth estimate rather than a decay estimate. Please state the range of k precisely.
  5. [Section 2.2, after Table 1] There is a typo 'statonary' that should be 'stationary'.
  6. [Section 3, equation (3.5)] The displayed identity contains a stray '??' before the equality sign; please remove it.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main asymptotic estimates are derived from the integral definition via stationary phase and van der Corput, and the only self-citation is for the definition and context, not for the load-bearing argument.

full rationale

The main derivation chain starts from the definition (1.7) of J_0^p and transforms it, by exact substitution, into the oscillatory integral representation (2.1); no target asymptotic estimate is used in that step. Theorem 1.4 then applies standard integration-by-parts and stationary-phase lemmas (Lemma 2.3, Lemma 2.5) to the explicit phases f_{p,phi} and g_{p,phi}, with the nondegeneracy of the second derivative at the stationary point and the vanishing of the amplitude at the endpoints obtained by direct computation. Theorem 1.5 applies van der Corput's lemma (Lemma 2.7) with k = 2/p. The delicate part is uniformity as delta tends to 0 near the axes. The paper proves Proposition 2.8 for this purpose, but the uniform estimate does not actually depend on that proposition: (2.8) already gives F_{p,delta}^{(2/p)}(theta) is nonzero on a fixed interval [0,a] for all 0 <= delta <= delta', and the proof re-selects delta' so that the delta-dependent stationary point pi/2 - theta_delta lies in [0,a]. By compactness the absolute value of the (2/p)-th derivative has a positive lower bound there, so Lemma 2.7 yields O(|eta|_p^{-p/2}) uniformly without needing Proposition 2.8. Thus Proposition 2.8 is not an input that is assumed to obtain the theorem; it is a supporting derivative estimate proved independently in Section 2.3. Section 3 explicitly labels the positive-order oscillatory integral representation as future work, so no unproved conjecture is used in the main proofs. The only self-citation, [10], supplies the definition of J_omega^p and the series representation used for context; these are definitions and background, not asymptotic conclusions imported as assumptions. No fitted parameter is renamed as a prediction, and no equation reduces to its own input by construction. Minor textual inconsistencies, such as the abstract writing '0 < p <= 1 or p = 2' versus the theorem statement '0 < p < 1 or p = 2', do not affect the derivation and are not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The main theorems use only standard asymptotic lemmas plus the exact oscillatory integral representation of J_0^p. The integrality condition 2/p in N for the uniform R^2 result is a domain hypothesis, and Proposition 2.8 is the paper's own technical lemma. No free parameters are fitted, and no new entities are postulated.

assumptions (5)
  • standard math Standard asymptotic lemmas: integration by parts (Lemma 2.3), stationary phase with endpoint contributions (Lemma 2.5), and van der Corput's lemma (Lemma 2.7) are valid as stated.
    Used throughout Section 2; cited from standard references [1], [2], [4], [16].
  • standard math The oscillatory integral representation (2.1) correctly represents J_0^p via the substitution t = cos^2 theta and trigonometric identities.
    Proposition 2.1 is derived exactly from definition (1.7) by a change of variables; no approximation is involved.
  • domain assumption For 0<p<1, psi[p](theta) = (cos theta sin theta)^(2/p - 1) is C^1 and vanishes at theta=0 and theta=pi/2, so the endpoint contributions from stationary phase are absent or negligible; for p=2, psi = 1.
    Required for Lemma 2.5 applicability at endpoints; discussed in Remark 2.6.
  • domain assumption The symmetry argument: it suffices to treat the first quadrant and the right neighborhood of the positive y-axis; the other quadrants follow by symmetry.
    Stated in the proofs of Theorems 1.4 and 1.5; the phase function signs in other quadrants require separate but similar treatment.
  • domain assumption For the uniform R^2 result, 2/p is a natural number (other than 1 and 2), ensuring the phase functions have C^{2/p} regularity and permitting the derivative lower bounds at the axes.
    This is the theorem's hypothesis for Theorem 1.5; without integrality, the phase lacks the required smoothness near the axes and the proof does not apply.

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Cite this review

Pith. "Pith review of Asymptotic evaluations of generalized Bessel function of order zero related to the p-circle lattice point problem." pith.science (2026). https://pith.science/paper/7XN3ZM5D

@misc{pith2026241110850,
  author       = {Pith},
  title        = {Pith review of: Asymptotic evaluations of generalized Bessel function of order zero related to the p-circle lattice point problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7XN3ZM5D}},
  note         = {Machine review of arXiv:2411.10850}
}
abstract

Let $p$ and $r$ be positive real numbers. Then, we consider the lattice point problem of the closed curve $p$-circle $\{x\in\mathbb{R}^{2}|\ |x_{1}|^{p}+|x_{2}|^{p}=r^{p}\}$ which is a generalization of the circle ($p=2$). Following the harmonic analytic approach of S. Kuratsubo and E. Nakai for the case of a circle, we need to investigate properties of appropriately generalized Bessel functions for $p$ in order to tackle the problem. Thus, in this paper, we derive asymptotic evaluations of the generalized Bessel function of order zero, such as uniformly asymptotic estimates on compact sets on quadrants of $\mathbb{R}^{2}$ for the cases $0<p\leq1$ or $p=2$, and, as stronger results, uniformly asymptotic estimates on $\mathbb{R}^{2}$ for the cases $p$ such that $\frac{2}{p}$ are the natural numbers.

Figures

Figures reproduced from arXiv: 2411.10850 by the authors.

Figure 1
Figure 1. Examples of the p-circle and the approximation by unit squares. On the other hand, in the cases p > 2, the following important theorem by E. Kr¨atzel is given by the representation ([11], (3.57)) Pp(r) = Ψ(r; p) + ∆(r; p) (1.3) decomposed by the second main term Ψ(r; p) ([11], (3.55)), which is represented as the series ex￾pansion consisting of the generalized Bessel functions ([11], Definition 3.3). It can be seen … view at source ↗

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