Omega Estimate for the Lattice Point Discrepancy of a Body of Revolution Using The Resonance Method

Nilmoni Karak

REVIEW 2 major objections 5 minor 15 references

Omega Estimate for the Lattice Point Discrepancy of a Body of Revolution Using The Resonance Method

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2506.22590 v1 pith:6KKG557Y submitted 2025-06-27 math.NT

Nilmoni Karak This is my paper

classification math.NT MSC 11P2111K3852C07

keywords OmegaboundlatticepointdiscrepancybodyofrevolutionresonancemethodtacfunctionBorelmeanvaluesquarefreeintegerslower

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 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.

What carries the argument

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 $\log_2 t$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $\lambda=\sqrt{2}$.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $B$.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Kamal,

This 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.

What'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.

The 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.

I 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.

Recommendation: 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.

Referee Report

2 major / 5 minor

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.

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 (2)

[Section 3, Eq. (3.4)] 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 2, Theorem 2.1] 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.

minor comments (5)

[Title and abstract] The title and abstract contain OCR-style typos such as 'ESTIMA TE', 'LA TTICE', and 'DISCREP ANC Y'; please correct them.
[Section 3, definition of A] 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'.
[Throughout] 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.
[Theorem 2.1 display] 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 3, after Eq. (3.5)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the omega estimate is a genuine deduction from an external resonance theorem; the unverified Q-linear independence is a correctness gap, not a circular reduction.

full rationale

The proof of Theorem 1.1 applies Mahatab's Theorem 2.1 to an explicitly constructed resonator set, then estimates the resulting sums with standard analytic number theory tools. The lower bound for S(t) in (3.3) is the theorem's lower bound for a cosine sum, and the main-term estimate (3.6) follows from Sathe's theorem, Stirling's formula, and a maximization over λ; λ = sqrt(2) is selected analytically, not fitted to the target omega bound. The Borel-mean identity (2.4) and Hafner's lemma convert the S(t) lower bound into the omega statement at the end, so the conclusion is not an ingredient of the construction. The central caveat is that the Q-linear independence hypothesis of Theorem 2.1 is stated as a hypothesis of the quoted theorem but is never verified for the set cM defined near (3.4); for the unit ball such independence can even fail. This is a serious correctness gap in applying the cited theorem, but it is not a circularity: no equation in the paper makes the claimed omega bound equal to the hypothesis, and the argument does not reduce by construction to its input. Therefore the circularity score is 0.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The ledger is dominated by borrowed tools: the resonance theorem (unpublished, advisor), the Borel mean lemma (Nowak), Sathe's count, and Hafner's integral lemma. The paper's own contribution is the two-variable resonator construction and the parameter balance, which are legitimate. The only hand-set quantities are lambda (a genuine maximizer) and the unverified Q-linear independence of M. There are no new particles, forces, or entities.

free parameters (5)

lambda (resonator exponent parameter) = sqrt(2)
Fixed at sqrt(2) to maximize the exponent of log_2 T in the main term; the maximum is genuine (derivative zero at log lambda = (log 2)/2), so this is an optimization parameter, not a fit.
alpha (resonance scale) = C^(-1)(log T)^(1/3)(log_2 T)^((1/3)(1 - lambda + lambda log lambda + lambda log 2))(log_3 T)^(1/6)
Balanced so that the resonator size |M| is of order log T, which makes the error terms in Theorem 2.1 negligible. A balancing parameter, standard for the resonance method.
C (large constant) = unspecified, taken large
Chosen large enough so e^(2M/C0) is much less than T^(1/4 - epsilon); harmless but load-bearing for the error term estimate.
A1, A2, A3, A4 = 2, 3/2, 1, 7/8
Exponents in Mahatab's Theorem 2.1, chosen to satisfy 0 < A4 < A3 < A2 < A1 and make the error terms small. Fixed choices, consistent with the hypotheses.
C0 = in (0,2), fixed
Constant from Theorem 2.1 defining the resonator interval [C0 alpha, 2 alpha]; not fitted by the paper.

assumptions (6)

domain assumption Mahatab's Theorem 2.1: for Q-linearly independent M subset of {lambda_n : C0 alpha <= lambda_n <= 2 alpha}, the maximum over t of the cosine sum is at least (pi/4e) times the sum over M of a_n, plus error terms.
Unproved in this paper; cited from the unpublished arXiv preprint [9] by the author's advisor. The entire lower bound for S(t) rests on it.
domain assumption Borel mean value asymptotic B(t) = -(1/(2 pi)) t S(t) + O(t^(3/4 + epsilon)) with the displayed S(t) and positive theta(m) (Lemma 2.2).
Quoted from Nowak [12, Eq. (13)] with s = 3. The proof also tacitly needs theta(m) bounded below to get f(l,m3) asymptotic to r(l), which the lemma does not state.
ad hoc to paper The constructed resonator set M is Q-linearly independent in the values H(sqrt(l),0,m3).
Stated as part of the construction but never verified. For the ball (H = ||u||) the box construction can yield repeated or Q-dependent values, so this is a genuine missing step.
standard math Sathe's theorem and Stirling's formula give the count of squarefree l with primes congruent to 1 mod 4 and omega(l) = [lambda log_2 alpha] up to powers of log_2 alpha and (log alpha)^(lambda - 1 - lambda log lambda - lambda log 2).
Standard analytic number theory (Tenenbaum II.6); the delicate point is the sign of the sqrt(log_2 alpha) factor, which the paper's text does not pin down cleanly.
standard math Hafner's Lemma 2.3.6 bounds the weighted integral in the contradiction step by C2 delta t F(t^2).
Published result used in the final contradiction argument.
standard math There are constants 0 < c1 < c2 and 0 < c3 < c4 with the box [c1,c2] x [c3,c4] inside the annulus between the level curves H = C0 and H = 2.
Geometric fact about smooth positive-curvature convex bodies; unproblematic.

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Pith. "Pith review of Omega Estimate for the Lattice Point Discrepancy of a Body of Revolution Using The Resonance Method." pith.science (2026). https://pith.science/paper/6KKG557Y

@misc{pith2026250622590,
  author       = {Pith},
  title        = {Pith review of: Omega Estimate for the Lattice Point Discrepancy of a Body of Revolution Using The Resonance Method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6KKG557Y}},
  note         = {Machine review of arXiv:2506.22590}
}

abstract

Using a recent method developed by Mahatab, we obtain an improved $\Omega$-bound for the error term arising in lattice counting problem of bodies of revolution in $\mathbb R^3$ around a coordinate axis and having smooth boundary with bounded nonzero curvature. This strengthens an earlier result by K\"uhleitner and Nowak.

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Aistleitner, K

C. Aistleitner, K. Mahatab and M. Munsch, Extreme values of the Riemann zeta function on the 1-line , Int. Math. Res. Not. 2019, no. 22, 6924–6932

work page 2019
[2]

Chamizo, Lattice points in bodies of revolution , Acta Arith

F. Chamizo, Lattice points in bodies of revolution , Acta Arith. 85 (1998), no. 3, 265–277

work page 1998
[3]

Guo, Lattice points in rotated convex domains , Rev

J. Guo, Lattice points in rotated convex domains , Rev. Mat. Iberoam. 31 (2015), no. 2, 411–438

work page 2015
[4]

J. L. Hafner, On the average order of a class of arithmetical functions , J. Number Theory 15 (1982), 36–76

work page 1982
[5]

Hlawka, ¨Uber Integrale auf konvexen K¨ orpern

E. Hlawka, ¨Uber Integrale auf konvexen K¨ orpern. I, Monatsh. Math. 54 (1950), 1–36

work page 1950
[6]

Kr¨ atzel, Lattice points, Mathematics and its Applications (East European Series), 33, Kluwer Acad

E. Kr¨ atzel, Lattice points, Mathematics and its Applications (East European Series), 33, Kluwer Acad. Publ., Dordrecht, 1988

work page 1988
[7]

K¨ uhleitner, Lattice points in bodies of revolution in R3: an Ω−-estimate for the error term , Arch

M. K¨ uhleitner, Lattice points in bodies of revolution in R3: an Ω−-estimate for the error term , Arch. Math. (Basel) 74 (2000), no. 3, 234–240

work page 2000
[8]

K¨ uhleitner and W

M. K¨ uhleitner and W. G. Nowak,The lattice point discrepancy of a body of revolution: improving the lower bound by Soundararajan ’s method,Arch. Math. (Basel) 83 (2004), 208–216

work page 2004

Show all 15 references

[9]

Mahatab, Omega Results for The Divisor and Circle Problems Using The Resonance Method , arXiv, 2025, arXiv: 2504.17032

K. Mahatab, Omega Results for The Divisor and Circle Problems Using The Resonance Method , arXiv, 2025, arXiv: 2504.17032

2025 arXiv
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G. Tenenbaum, Introduction to analytic and probabilistic number theory , Graduate Studies in Mathematics 163, Amer. Math. Soc. 2015 THE LATTICE POINT DISCREPANCY OF A BODY OF REVOLUTION 7 Nilmoni Karak, Department of Mathematics, Indian Institute of Technology Kharagpur, Khara...

2015

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Aistleitner, K

C. Aistleitner, K. Mahatab and M. Munsch, Extreme values of the Riemann zeta function on the 1-line , Int. Math. Res. Not. 2019, no. 22, 6924–6932

work page 2019

[2] [2]

Chamizo, Lattice points in bodies of revolution , Acta Arith

F. Chamizo, Lattice points in bodies of revolution , Acta Arith. 85 (1998), no. 3, 265–277

work page 1998

[3] [3]

Guo, Lattice points in rotated convex domains , Rev

J. Guo, Lattice points in rotated convex domains , Rev. Mat. Iberoam. 31 (2015), no. 2, 411–438

work page 2015

[4] [4]

J. L. Hafner, On the average order of a class of arithmetical functions , J. Number Theory 15 (1982), 36–76

work page 1982

[5] [5]

Hlawka, ¨Uber Integrale auf konvexen K¨ orpern

E. Hlawka, ¨Uber Integrale auf konvexen K¨ orpern. I, Monatsh. Math. 54 (1950), 1–36

work page 1950

[6] [6]

Kr¨ atzel, Lattice points, Mathematics and its Applications (East European Series), 33, Kluwer Acad

E. Kr¨ atzel, Lattice points, Mathematics and its Applications (East European Series), 33, Kluwer Acad. Publ., Dordrecht, 1988

work page 1988

[7] [7]

K¨ uhleitner, Lattice points in bodies of revolution in R3: an Ω−-estimate for the error term , Arch

M. K¨ uhleitner, Lattice points in bodies of revolution in R3: an Ω−-estimate for the error term , Arch. Math. (Basel) 74 (2000), no. 3, 234–240

work page 2000

[8] [8]

K¨ uhleitner and W

M. K¨ uhleitner and W. G. Nowak,The lattice point discrepancy of a body of revolution: improving the lower bound by Soundararajan ’s method,Arch. Math. (Basel) 83 (2004), 208–216

work page 2004