{"id":"9b6c3566-40f1-49f2-a954-305da31ef999","arxiv_id":"2506.22590","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"For smooth convex bodies of revolution in R^3, the lattice point discrepancy satisfies P_B(t) = Omega_-(t^(1/2)(log t)^(1/3)(log_2 t)^((2/3)(sqrt(2)-1))(log_3 t)^(-1/3)), improving the prior exponent of log_3 t from -2/3 to -1/3.","lead":"This paper proves a sharper lower bound on how badly the count of integer grid points inside a large smooth 3D shape of revolution can deviate from the smooth volume estimate. It is a small step in a classic number theory problem, and its proof rests on a recently introduced resonance method together with one unverified independence condition.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Q-linear independence of the resonator frequencies is never verified and is impossible for the unit ball: |M| is ≫α^3 while all frequencies sqrt(l+m3^2) live in a Q-span of dimension O(α^2).","rationale":"The reader's verdict is CONDITIONAL, and I agree. The central construction is otherwise internally consistent: the choice λ=√2 and the scale α=(log T)^{1/3} reproduce the known log_2 exponent, the Sathe counting is plausible, and the error estimates numerically suppress the O-terms. The single place where an external hypothesis must carry the argument is the application of Mahatab's Theorem 2.1 to the set \\hat M. The theorem is quoted from an unpublished preprint and its independence hypothesis is not verified. My stress-test sharpens this: for the ball, the hypothesis is not merely unproved but false, because the number of lattice pairs in \\hat M exceeds the dimension of the Q-vector space generated by the corresponding frequencies. This is a load-bearing defect in the proof as written, not just a missing citation check. However, this does not establish that Theorem 1.1 is false; it shows the method in this paper does not cover a natural special case. A revised version could restrict the class of bodies or find a different way to control collisions. Hence the appropriate verdict remains CONDITIONAL, and the reader's stated condition should be made explicit: verify or bypass Q-linear independence.","tokens_in":6047,"tokens_out":21645,"duration_ms":249609,"concrete_test":"Set B to the unit ball and fix the constants c1,c2,c3,c4 from Section 3. Let N = (c2^2+c4^2)α^2 and let D(N) = #{n≤N : n squarefree}. Compute the resonator size M(α) = |\\hat M| from (3.4)-(3.5) and compare M(α) with D(N). Since D(N) ∼ (6/π^2)N and M(α) ≍ α^3/(log α)^c, one gets M(α)/D(N) ≍ α/(log α)^c → ∞. This counting comparison settles that the Q-linear independence hypothesis of Theorem 2.1 cannot hold for the ball for large α, so the quoted theorem cannot be invoked in the proof of Theorem 1.1 without a different resonator or a modified claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3 applies Theorem 2.1 to the resonator set \\hat M defined around (3.4). The theorem requires the frequencies λ_n = H(√l, 0, m3) attached to \\hat M to be linearly independent over Q. The paper states this as part of the quoted theorem but never verifies it. This is decisive: without independence, the lower bound (3.3) for S(t) is not justified. For the unit ball B, H(u)=||u||, so λ_n = √(l+m3^2). Every such number is √n for n = l+m3^2 ≤ (c2^2+c4^2)α^2. The Q-vector space spanned by {√n : n ≤ N} has dimension equal to the number of squarefree n ≤ N, about (6/π^2)N, so any Q-linearly independent set of these frequencies has size O(α^2). But the constructed \\hat M has size |\\hat M| ≍ α^3/(log α)^c for some c>0, which for large α is much larger than α^2. Consequently no subset of \\hat M of this cardinality can be Q-linearly independent; many pairs even produce equal values (at small scale 14 = 13 + 1^2 = 5 + 3^2). Since the unit ball is a compact C∞ rotational body with positive curvature, it lies in the scope of Theorem 1.1. The same obstruction affects bodies with algebraic support functions. The proof must either find a Q-independent resonator retaining the large cardinality, which is impossible in the ball case, or supply a different argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims an improved Omega-lower bound for the lattice point discrepancy P_B(t) of a compact convex body of revolution in R^3 with C^infinity boundary and positive curvature. The claimed bound is t^{1/2}(log t)^{1/3}(log_2 t)^{(2/3)(sqrt(2)-1)}(log_3 t)^{-1/3}, improving the exponent of log_3 t from -2/3 to -1/3 relative to the Kuehleitner-Nowak bound. The proof applies a resonance theorem of Mahatab to a constructed set of frequencies H(sqrt(l),0,m3), with l squarefree, all prime factors congruent to 1 mod 4, and m3 in a dyadic interval, then balances parameters and converts the lower bound on the Borel mean value into an Omega result via a contradiction argument.","tokens_in":6236,"tokens_out":13627,"duration_ms":139859,"significance":"The claimed improvement is meaningful and the parameter bookkeeping is coherent: the optimizer lambda = sqrt(2) is a genuine maximizer of the displayed exponent and reproduces the known log_2 exponent, and the choice of alpha makes the resonator cardinality of order log T, matching the intended error control. The Borel-mean reduction and the final contradiction are standard. However, the proof depends entirely on applying Mahatab's Theorem 2.1 to a set M that must be Q-linearly independent, and the paper supplies no verification; for the unit ball the constructed set is far too large to be Q-linearly independent. Since the unit ball is within the theorem's scope, this is a load-bearing gap that invalidates the proof as it stands.","major_comments":[{"comment":"The set \\hat M is asserted to be the resonator for Theorem 2.1, but Theorem 2.1 requires the frequencies lambda_n = H(sqrt(l),0,m3) attached to M to be linearly independent over Q. No argument is given. For the unit ball, H(u)=||u||, so every such frequency is sqrt(l+m3^2) and lies in the Q-span of {sqrt(d): d squarefree, d <= D} with D <= (c2^2+c4^2) alpha^2; this span has dimension asymptotically D = O(alpha^2). The construction gives |\\hat M| asymptotically alpha^3/(sqrt(log_2 alpha)(log alpha)^{1.055}) for lambda = sqrt(2), which is much larger than alpha^2 for large alpha. Even after the balancing alpha asymptotically (log T)^{1/3}, the required |M| of order log T exceeds the maximum possible size O((log T)^{2/3}) of any Q-independent subset for the ball. Thus the hypothesis of the quoted theorem is not merely unverified; it is false for an in-scope body, and the proof of the key lower bound (3.3) collapses.","section":"Section 3, Eq. (3.4)"},{"comment":"The main lower bound rests on a theorem quoted from an unpublished arXiv preprint (arXiv:2504.17032), and the statement as reproduced is ambiguous: M is used both as a set of indices in sum_{n in M} a_n and as a set of values lambda_n said to be linearly independent over Q. The manuscript should either prove the theorem in an appendix, state the precise meaning of linear independence in this context, or cite a published version; in the present form the central inequality (3.3) is not independently checkable from the manuscript.","section":"Section 2, Theorem 2.1"}],"minor_comments":[{"comment":"The title and abstract contain OCR-style typos such as 'ESTIMA TE', 'LA TTICE', and 'DISCREP ANC Y'; please correct them.","section":"Title and abstract"},{"comment":"The set A is defined awkwardly as '{q in N : p congruent to 1 mod 4, if p|q and p is prime; and q is square free}'; rewrite this as 'q squarefree and p congruent to 1 mod 4 for every prime p dividing q'.","section":"Section 3, definition of A"},{"comment":"The symbol M is used both for the set of indices and for its cardinality (M = |M|); please use distinct notation, for example \\mathcal{M} for the set and M for its cardinality.","section":"Throughout"},{"comment":"The displayed range of t in Theorem 2.1 is malformed ('T A3 /2 <= t <= 2A2 2T A2 log2 T'); the intended formula needs correction.","section":"Theorem 2.1 display"},{"comment":"The phrase 'using the assumption XH(...)^2 << 1' is confusing because the condition is verified later; say explicitly that the verification follows from the choice of alpha made in the next step.","section":"Section 3, after Eq. (3.5)"}],"recommendation":"reject","confidential_remarks":"The independence gap is decisive: the unit ball is an in-scope body for which the constructed resonator cannot satisfy the stated hypothesis of Mahatab's theorem. I see no local repair within the method as presented, so I recommend rejection, despite the plausibility of the claimed numerical improvement."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Kamal,\n\nThis is a short note on Karak's arXiv:2506.22590. The headline: the paper claims the best known Omega lower bound for the lattice point discrepancy of smooth bodies of revolution in R^3, improving the log_3 exponent from -2/3 to -1/3 by importing Mahatab's resonance method. The exposition is clean and the optimization is genuinely interesting: choosing λ=√2 exactly recovers the known log_2 exponent, which suggests the machinery is doing the right thing.\n\nWhat's new: this is the first time the resonance method is applied to this lattice point problem. The construction of the resonator set \\hat M from pairs (l,m3) with l squarefree, all prime factors 1 mod 4, and counting by Sathe's theorem is neat. The parameter balance checks out, and the improvement is structurally exactly what the method should buy over Soundararajan's approach. If the theorem is true, it's a solid step.\n\nThe soft spot is not minor. The application of Mahatab's Theorem 2.1 requires the frequencies H(√l,0,m3) for the chosen resonator to be linearly independent over Q. The paper states this as part of the quoted theorem but never verifies it for \\hat M. For the unit ball, which is a perfectly valid body in the theorem, H(√l,0,m3) = √(l+m3^2). The set \\hat M has ~α^3 pairs, but all these numbers lie in the Q-span of {√n: n≤cα^2}, whose dimension is ~α^2. So a linearly independent subset of \\hat M can have at most O(α^2) elements. The paper's main term (3.5) uses all ~α^3 pairs; restricting to a Q-independent subset loses a factor of α and the final Omega bound collapses. There is also the issue that the key resonator theorem is from an unpublished preprint (Mahatab, arXiv:2504.17032), so the foundation is not independently checked.\n\nI should say the argument is not obviously a dead end. For a generic body, the values H(√l,0,m3) may well be algebraically independent, and the proof could go through for a restricted class. But as stated, Theorem 1.1 covers every smooth body of revolution, including the ball, and for the ball the proof fails. The author needs to either prove independence for a class of bodies, or change the resonator to respect the dimension bound.\n\nRecommendation: send to a serious referee, but expect major revision. The idea is worth engaging with, and an expert referee could help determine whether the method can be salvaged. I would not cite this as it stands.","headline":"The claimed log_3 t improvement is genuine in spirit, but the resonator's Q-linear independence fails for the unit ball, so Theorem 1.1 is unproved as stated.","tokens_in":7016,"tokens_out":8351,"would_cite":false,"duration_ms":78569,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11P21","11K38","52C07"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper derives a new $\\Omega_-$ lower bound for the lattice point discrepancy of smooth bodies of revolution in $\\mathbb{R}^3$, improving the power of $\\log_3 t$ in the previous best bound from $-2/3$ to $-1/3$ by applying a resonance…","keywords":["Omega bound","lattice point discrepancy","body of revolution","resonance method","tac function","Borel mean value","squarefree integers","lower bound"],"falsifier":"For the unit ball $B$, where $H(u)=\\|u\\|$, the independence hypothesis applied to the resonator set requires the numbers $\\sqrt{l+m_3^2}$ for all chosen pairs $(l,m_3)$ to be linearly independent over $\\mathbb{Q}$. A direct search for two distinct pairs in the resonator set with the same value of $l+m_3^2$ would exhibit equal frequencies and hence a $\\mathbb{Q}$-linear relation; if such a pair exists for moderately large $\\alpha$, the application of Theorem 2.1 in this paper cannot be valid.","tokens_in":5605,"feed_emoji":"📐","tokens_out":14673,"duration_ms":127775,"temperature":0.7,"pith_summary":"This paper claims a sharper $\\Omega_-$ bound for the lattice point discrepancy of large homothetic copies of a smooth, rotation-invariant convex body in $\\mathbb{R}^3$. The discrepancy $P_B(t)$ counts how much the number of integer lattice points inside $\\sqrt{t}B$ differs from the volume term $\\operatorname{vol}(B)t^{3/2}$. The author shows that, assuming the resonance theorem stated as Theorem 2.1, $P_B(t)=\\Omega_-\\!\\left(t^{1/2}(\\log t)^{1/3}(\\log_2 t)^{\\frac{2}{3}(\\sqrt{2}-1)}(\\log_3 t)^{-1/3}\\right)$, meaning the discrepancy is at least a constant times this size in the negative direction for infinitely many $t$. This improves the exponent of the third iterated logarithm from $-2/3$ to $-1/3$ in the earlier bound (1.2). The proof goes through a Borel mean value identity that reduces the discrepancy to an exponential sum, which is then forced to be large by a carefully chosen resonator set.","feed_headline":"Lattice-point discrepancy bound sharpened for bodies of revolution","feed_subtitle":"The power of the third iterated log improves from -2/3 to -1/3 in the Omega lower bound.","key_machinery":"The machine is the tac function $H(u)=\\max_{v\\in B}\\langle u,v\\rangle$, a positive homogeneous support-type function which, for bodies of revolution, reduces to $H(u_1,u_2,u_3)=H(\\sqrt{u_1^2+u_2^2},0,u_3)$. Pairing each lattice point with $(l,m_3)$, where $l=m_1^2+m_2^2$, gives frequencies $\\lambda_n=H(\\sqrt{l},0,m_3)$, and these are fed into the resonance theorem (Theorem 2.1) to produce a large value of the exponential sum $S(t)$ on a short interval of $t$. The resonator set consists of pairs $(l,m_3)$ with $l$ squarefree, all prime factors congruent to $1\\bmod 4$, exactly $\\lceil\\lambda\\log_2\\alpha\\rceil$ such factors, and with $l$ and $m_3$ in intervals of length comparable to $\\alpha$; an asymptotic count of such integers sizes the set, and $\\lambda=\\sqrt{2}$ is chosen to maximize the resulting exponent of $\\log_2 t$.","core_discovery":"The central claim is that for every compact convex body $B\\subset\\mathbb{R}^3$ that contains the origin, has $C^{\\infty}$ boundary with positive bounded curvature, and is invariant under rotations about a coordinate axis, the lattice point discrepancy obeys $P_B(t)=\\Omega_-\\!\\left(t^{1/2}(\\log t)^{1/3}(\\log_2 t)^{\\frac{2}{3}(\\sqrt{2}-1)}(\\log_3 t)^{-1/3}\\right)$. Here $\\Omega_-$ means there are arbitrarily large $t$ for which $P_B(t)$ is at most a negative constant times the displayed size. This supersedes the bound (1.2), whose power of $\\log_3 t$ is $-2/3$. The argument reduces the discrepancy to the Borel mean value $B(t)=-\\frac{1}{2\\pi}tS(t)+O(t^{3/4+\\epsilon})$, with $S(t)$ an exponential sum over the tac function $H$, and then uses a resonator set of pairs $(l,m_3)$ to force $S(t)$ to be large; the optimal parameter choice is $\\lambda=\\sqrt{2}$.","pith_inferences":["A concrete test of the argument would be to check whether the constructed frequencies $\\{\\lambda_n\\}$ are actually $\\mathbb{Q}$-linearly independent; for the Euclidean ball this reduces to checking whether any two pairs $(l,m_3)$ in the resonator set satisfy $l+m_3^2=l'+m_3'^2$, which a finite computer search could settle.","If the independence hypothesis fails for some admissible body, the resonance bound (3.6) may still hold but would need a different proof; the present paper does not address this.","The method's success here suggests that similar exponent improvements might be available for lattice point discrepancy in higher-dimensional bodies of revolution, as long as an analogue of the reduction (2.2) and a suitable resonance theorem exist.","Because Theorem 2.1 is imported from an unpublished preprint, the unconditional status of the result depends entirely on that preprint; until it appears, Theorem 1.1 should be read as conditional."],"forward_implications":["For infinitely many large scales $t$, the lattice point discrepancy of any admissible body of revolution is negative and at least a constant times $t^{1/2}(\\log t)^{1/3}(\\log_2 t)^{\\frac{2}{3}(\\sqrt{2}-1)}(\\log_3 t)^{-1/3}$.","This improves the previously known $\\Omega_-$ bound (1.2) by a factor of $(\\log_3 t)^{1/3}$.","The resonance method, already used for circle and divisor problems, transfers to lattice point discrepancy via the tac function and the Borel mean value reduction.","The lower bound applies uniformly to all bodies satisfying the smoothness and rotational symmetry conditions, not only to the Euclidean ball.","The optimal choice $\\lambda=\\sqrt{2}$ in the resonator construction is determined by a quadratic optimization and is independent of the particular body $B$."],"supporting_citations":[{"why":"Supplies Theorem 2.1, the resonance theorem that produces the large lower bound for the exponential sum $S(t)$ from which the $\\Omega_-$ result follows.","marker":"[9]"},{"why":"Provides the earlier bound (1.2) that Theorem 1.1 improves, and its Section 2 informs the Borel mean value setup.","marker":"[8]"},{"why":"Supplies the asymptotic count of squarefree integers with a prescribed number of prime factors, used to size the resonator set.","marker":"[13]"},{"why":"Cited together with the previous item for the same counting estimate and Stirling's formula.","marker":"[15]"},{"why":"Provides Eq. (13) from which Lemma 2.2, the Borel mean value asymptotic formula, is obtained.","marker":"[12]"},{"why":"Supplies Lemma 2.3.6, used to estimate the integral in the final contradiction argument.","marker":"[4]"},{"why":"Underlies the method that produced the previous best bound (1.2), the baseline this paper strengthens.","marker":"[14]"}],"fun_headline_variants":["Resonance method tightens Omega bound for lattice discrepancy","Improved Omega lower bound for revolution lattice discrepancy","Third log exponent sharpened in Omega discrepancy bound","Resonance method improves lattice discrepancy Omega bound"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The result stands on a resonance theorem quoted from an unpublished preprint and on the unverified hypothesis that the constructed frequencies $H(\\sqrt{l},0,m_3)$ are linearly independent over $\\mathbb{Q}$; if either fails, the stated bound does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Resonance method tightens Omega bound for lattice discrepancy","Improved Omega lower bound for revolution lattice discrepancy","Third log exponent sharpened in Omega discrepancy bound","Resonance method improves lattice discrepancy Omega bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001235,"raw_usage":{"total_tokens":5014,"prompt_tokens":832,"completion_tokens":4182,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":448,"completion_tokens_details":{"reasoning_tokens":4122}},"tokens_in":448,"tokens_out":4182,"duration_ms":28218,"temperature":1.0,"reasoning_tokens":4122,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:08:49.870470+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the unit ball $B$, where $H(u)=\\|u\\|$, the independence hypothesis applied to the resonator set requires the numbers $\\sqrt{l+m_3^2}$ for all chosen pairs $(l,m_3)$ to be linearly independent over $\\mathbb{Q}$. A direct search for two distinct pairs in the resonator set with the same value of $l+m_3^2$ would exhibit equal frequencies and hence a $\\mathbb{Q}$-linear relation; if such a pair exists for moderately large $\\alpha$, the application of Theorem 2.1 in this paper cannot be valid.","supporting_citations":[{"cited_title":"Omega Results for The Divisor and Circle Problems Using The Resonance Method","cited_arxiv_id":"2504.17032","evidence_quote":"Supplies Theorem 2.1, the resonance theorem that produces the large lower bound for the exponential sum $S(t)$ from which the $\\Omega_-$ result follows."},{"cited_title":"K¨ uhleitner and W","cited_arxiv_id":null,"evidence_quote":"Provides the earlier bound (1.2) that Theorem 1.1 improves, and its Section 2 informs the Borel mean value setup."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the asymptotic count of squarefree integers with a prescribed number of prime factors, used to size the resonator set."},{"cited_title":"Tenenbaum, Introduction to analytic and probabilistic number theory , Graduate Studies in Mathematics 163, Amer","cited_arxiv_id":null,"evidence_quote":"Cited together with the previous item for the same counting estimate and Stirling's formula."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides Eq. (13) from which Lemma 2.2, the Borel mean value asymptotic formula, is obtained."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 2.3.6, used to estimate the integral in the final contradiction argument."},{"cited_title":"Soundararajan, Omega results for the divisor and circle problems, Int","cited_arxiv_id":null,"evidence_quote":"Underlies the method that produced the previous best bound (1.2), the baseline this paper strengthens."}],"review_version":1}