{"id":"148e8016-6383-460f-87f9-496e486e0666","arxiv_id":"2411.10850","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the p-circle generalized Bessel function J_0^p, the paper proves uniform O(|eta|^{-p/2}) decay on R^2 when 2/p is an integer greater than 2, and O(|eta|^{-1/2}) decay on compact sectors for all 0<p<1 and p=2.","lead":"Kitajima proves new decay estimates for a generalized Bessel function built from the p-norm, the key analytic object in the p-circle lattice point problem. The bounds are uniform over the whole plane for the family p=2/n, giving a concrete tool for attacking a long-open counting problem.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Theorem 1.5's uniform decay is supported by a fixed-interval lower bound (2.8), and Proposition 2.8 is not load-bearing.","rationale":"I read the proof of Theorem 1.5 and traced the role of Proposition 2.8. The proof after re-selecting δ′ places the δ-dependent stationary point inside [0,a], where (2.8) already guarantees F^{(2/p)} is nonzero; van der Corput's lemma then gives the O(|η|^{-p/2}) bound uniformly in δ, since nonzero on a compact rectangle implies a uniform positive lower bound. Thus Proposition 2.8 is not essential to the central claim. I also spot-checked Proposition 2.8's O,Omega(1) claim for n=3,4,5 by Taylor expansion and found it consistent. The remaining issues are cosmetic: the abstract includes p=1 while Theorem 1.4 and Remark 2.6 exclude it, and the abstract's 'natural numbers' should say 'other than 2'. These do not change the correctness of the asymptotic estimate. The reader's conditional accept is reasonable; no adjustment is needed.","tokens_in":18621,"tokens_out":26978,"duration_ms":250025,"concrete_test":"Compute F_{p,δ}^{(2/p)}(θ) explicitly for p=2/3 (n=3) and p=1/2 (n=4), and evaluate its minimum absolute value on [0,a]×[0,δ′] for, say, a=0.1 and δ′=0.01; if the minimum is bounded away from 0, the uniform lower bound (2.8) used in Theorem 1.5 is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing concern identified. The reader's flagged assumption is Proposition 2.8, but the needed uniform lower bound for van der Corput is already supplied by (2.8): F_{p,δ}^{(2/p)}(θ) is nonzero on a fixed interval [0,a] for 0≤δ≤δ′, and by compactness its absolute value is bounded below by a positive constant. After re-selecting δ′ so the δ-dependent stationary point π/2−θδ lies in [0,a], Lemma 2.7 on [0,a] yields I_{f,±}=O(|η|_p^{−p/2}) uniformly without invoking Proposition 2.8. The proposition's pointwise O,Omega(1) is consistent with direct Taylor checks for n=3,4,5. Minor abstract/statement inconsistencies (0<p≤1 vs 0<p<1; natural numbers 'other than 2') do not affect the central claim.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":18726,"tokens_out":16659,"duration_ms":163797,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"Abstract / Theorem 1.4"},{"comment":"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}, ...'.","section":"Theorem 1.5"},{"comment":"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.","section":"Section 2.2, proof of Theorem 1.5"},{"comment":"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.","section":"Section 2.3, equation (2.15)"},{"comment":"There is a typo 'statonary' that should be 'stationary'.","section":"Section 2.2, after Table 1"},{"comment":"The displayed identity contains a stray '??' before the equality sign; please remove it.","section":"Section 3, equation (3.5)"}],"recommendation":"minor_revision","confidential_remarks":"This is a specialized but sound contribution. The central estimates have plausible, standard proofs, and the one flagged delicate point (Proposition 2.8) is not actually load-bearing for Theorem 1.5 because of the uniform lower bound in (2.8). The main issues are presentation-level: the abstract/statement inconsistencies about p=1 and p=2, and the very dense proof of Proposition 2.8 which could be streamlined or better explained. I recommend minor revision rather than major revision, as no load-bearing mathematical error was identified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a genuine, useful technical contribution. For 0<p<1, nobody had uniform asymptotics for the order-zero generalized Bessel function J_0^p, and for 2/p an integer the paper gives the first uniform-on-R^2 bounds. The connection to the p-circle problem is honestly scoped: this is a stepping stone, not a solution.\n\nWhat is new: the oscillatory integral representation (Proposition 2.1) and the axis-neighborhood analysis with the delta-dependent stationary point. The main estimates follow from the integral definition via standard van der Corput and stationary phase arguments--no fitted parameters, no circularity. The paper also correctly says where the method stops: Remark 2.6 for 1<p<2 and p>2, and Section 3 for positive orders.\n\nSoft spots are mostly presentation. Theorem 1.5 is worded confusingly: 'natural numbers other than 2' should mean 2/p in N\\{2}, but then p=2 (which gives 2/p=1) is listed separately. The v4 abstract says 0<p<1, so the reader's note about 0<p<=1 is probably from an earlier version. The deeper concern flagged by the reader--Proposition 2.8--is real in that the proof is long and dense, but it does not actually carry the theorem. The uniform R^2 bound follows from the fixed-interval lower bound (2.8) alone, because the moving stationary point lies inside [0,a] for small delta, and on that interval the (2/p)-th derivative is bounded below independently of delta. The proposition is extra, not load-bearing. I did not machine-check its expansion, but the central argument stands without it.\n\nWho this is for: people working on lattice point problems or generalized Bessel functions. It deserves a serious referee; with Theorem 1.5 reworded and Proposition 2.8 either simplified or marked optional, it is a conditional accept. I would take it to a reading group if the group cares about oscillatory integrals.","headline":"Genuinely new uniform asymptotics for a generalized Bessel function; the core proof is sound but the paper overstates the role of its densest proposition.","tokens_in":19333,"tokens_out":12950,"would_cite":false,"duration_ms":111435,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11P21","33C10","42B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $2/p\\in\\mathbb{N}\\setminus\\{1,2\\}$.","keywords":["lattice point problem","p-circle","superellipse","generalized Bessel function","uniform asymptotic estimates","oscillatory integral","van der Corput lemma"],"falsifier":"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.","tokens_in":18354,"feed_emoji":"🧮","tokens_out":13074,"duration_ms":110623,"temperature":0.7,"pith_summary":"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 $0<p<1$ and for $p=2$, matching the classical decay of the ordinary Bessel function. The main new result is the whole-plane estimate for the countably infinite family $p=2/N$ with $N\\ge3$: there the uniform decay $J_0^p(\\eta)=O(|\\eta|_p^{-p/2})$ holds in every direction, including arbitrarily close to the axes. These uniform bounds are exactly the ingredient that the generalized series-expansion method needs in order to transfer the circle's lattice-point argument to unsolved superellipse cases.","feed_headline":"Bessel function decays uniformly in all directions for p-circles","feed_subtitle":"When 2/p is a whole number, this Bessel function decays by a single power law everywhere, helping count lattice points in p-circles.","key_machinery":"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.","core_discovery":"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 $J_0(|x|)=O(|x|^{-1/2})$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"defines the generalized Bessel functions and supplies the series expansion that motivates studying their asymptotics.","marker":"[10]"},{"why":"supplies the circle-case harmonic-analytic expansion and the error-term program being generalized to arbitrary p.","marker":"[13]"},{"why":"provides the lattice-point decomposition for p>2 in which generalized Bessel functions appear as main-term series.","marker":"[11]"},{"why":"is the source of the stationary-phase formulas used for the compact-sector estimate.","marker":"[2]"},{"why":"is the source of van der Corput's lemma used for the uniform whole-plane estimate.","marker":"[4]"},{"why":"gives the oscillatory-integral tools and the Bessel integral representation used for the p=2 case.","marker":"[16]"},{"why":"provides the classical Bessel asymptotics against which the p=2 rate is checked.","marker":"[17]"},{"why":"documents an earlier independent study of the same order-zero generalized Bessel function outside number theory.","marker":"[3]"}],"fun_headline_variants":["Uniform Bessel decay aids p-circle lattice point counting","p-circle Bessel function gets uniform decay bounds","When 2/p is natural, Bessel decay is uniform on R^2","Uniform asymptotic estimates for generalized Bessel functions","Bessel decay uniformly for p-circles when 2/p is integer"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Uniform Bessel decay aids p-circle lattice point counting","p-circle Bessel function gets uniform decay bounds","When 2/p is natural, Bessel decay is uniform on R^2","Uniform asymptotic estimates for generalized Bessel functions","Bessel decay uniformly for p-circles when 2/p is integer"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000511,"raw_usage":{"total_tokens":2498,"prompt_tokens":968,"completion_tokens":1530,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":584,"completion_tokens_details":{"reasoning_tokens":1446}},"tokens_in":584,"tokens_out":1530,"duration_ms":11876,"temperature":1.0,"reasoning_tokens":1446,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T19:15:22.293900+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Kuratsubo & E","cited_arxiv_id":null,"evidence_quote":"supplies the circle-case harmonic-analytic expansion and the error-term program being generalized to arbitrary p."},{"cited_title":"Kr¨ atzel, Lattice Points, Kluwer Academic Publication, 1988","cited_arxiv_id":null,"evidence_quote":"provides the lattice-point decomposition for p>2 in which generalized Bessel functions appear as main-term series."},{"cited_title":"Bleistein & R.A","cited_arxiv_id":null,"evidence_quote":"is the source of the stationary-phase formulas used for the compact-sector estimate."},{"cited_title":"Duoandikoetxea ; translated and revised by D","cited_arxiv_id":null,"evidence_quote":"is the source of van der Corput's lemma used for the uniform whole-plane estimate."},{"cited_title":"Stein with the assistance of T.S","cited_arxiv_id":null,"evidence_quote":"gives the oscillatory-integral tools and the Bessel integral representation used for the p=2 case."},{"cited_title":"Watson, A treatise on the theory of Bessel functions, 2nd ed., Cambridge Univ","cited_arxiv_id":null,"evidence_quote":"provides the classical Bessel asymptotics against which the p=2 rate is checked."},{"cited_title":"zu Castell, Generalized Bessel functions for p-radial functions, Constr","cited_arxiv_id":null,"evidence_quote":"documents an earlier independent study of the same order-zero generalized Bessel function outside number theory."}],"review_version":1}