Galois orbits of torsion points over polytopes near atoral sets

Chenying Lin

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Galois orbits of torsion points over polytopes near atoral sets

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper proves that for essentially atoral Laurent polynomials, Galois-averaged sums of $\log|P|$ over torsion-point conjugates whose arguments land in any polytope converge to the Lebesgue integral over that polytope, with error…

desk verdict The polytope equidistribution theorem is a real extension of Dmitrov-Habegger and the proof looks coherent, but the explicit exponent advertised in Proposition 1.2 is not supported by Algorithm 1 as printed; the numerical claim needs fixing before the paper is publishable. read the letter →

arxiv 2412.11156 v1 pith:SD5MVCFV submitted 2024-12-15 math.NT math.CO

Chenying Lin This is my paper · ORCID

classification math.NTmath.CO MSC 11J8311G5014G4037P3052B11

keywords GaloisequidistributiontorsionpointsstrictnessdegreecotropicalizationatoralLaurentpolynomialsKoksmainequalityMahlermeasureheightsinprojectivespace

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 establishes a quantitative equidistribution theorem for the logarithmic absolute value of essentially atoral Laurent polynomials on polytope-restricted Galois orbits of torsion points of $\mathbb G_m^d$. For a $d$-dimensional polytope $\Delta\subset[0,1)^d$, it proves that the average of $\log|P(\omega^\sigma)|$ over conjugates whose arguments fall in $\Delta$ differs from the Lebesgue integral $\int_\Delta\log|P(e(x))|\,dx$ by at most a constant times $\delta(\omega)^{-\kappa}$, where $\delta(\omega)$ is the strictness degree. This makes the full-cube result of [DH24] the special case $\Delta=[0,1)^d$ and adds an explicit algorithm for the exponent $\kappa(d,k)$. The proof works by deriving a Koksma inequality over polytopes, using a continuous approximation of the characteristic function of $\Delta$. As an application, the height of a projective intersection point in a two-dimensional example converges to $2\zeta(3)/(3\zeta(2))$ with the explicit rate $\delta(\omega)^{-1/(261\cdot 5^5)}$, answering a question raised in [GS23].

What carries the argument

The central object is a piecewise-affine continuous characteristic function $\chi^c_{\Delta,\epsilon}$ of the polytope $\Delta$. It is built by shrinking $\Delta$ toward the center of an inscribed cubic ball by the factor $1-\epsilon$, setting the function to $1$ on the shrunk polytope, $0$ outside $\Delta$, and interpolating linearly across the truncated hyperpyramids between each facet and its shrunk copy. Proposition 3.15 bounds its modulus of continuity by $t/(\epsilon\,\mathrm{inrad}(\Delta))$; plugging this into the discrepancy estimate gives Theorem 3.18, a Koksma inequality for polytopes that controls the difference between a discrete average over points in $\Delta$ and the integral of a continuous function over $\Delta$. Around the logarithmic singularities of $\log|P|$, the bounded-log truncation $\log_r$ and the counting and local-volume estimates imported from the full-cube proof absorb the remaining errors.

What would settle it

For a fixed skew triangle $\Delta$ in $[0,1)^2$ and $P(T_1,T_2)=T_1-1$, compute the left side of Theorem 1.1 for a sequence of torsion points with $\delta(\omega)\to\infty$; if the error decays slower than any fixed power of $\delta(\omega)^{-1}$, the theorem fails. A more local check: take $\epsilon=10^{-3}$ and two points $x,y$ at distance $t=10^{-1}$ lying in two different truncated hyperpyramids of $\Delta$; if $|\chi^c_{\Delta,\epsilon}(x)-\chi^c_{\Delta,\epsilon}(y)|>t/(\epsilon\,\mathrm{inrad}(\Delta))$, then Proposition 3.15 is false.

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Extended reading notes

Core claim

The central claim is that equidistribution of $\log|P|$ along Galois orbits survives restriction to the cotropical preimage of any polytope, with a power-law rate. Precisely, for an essentially atoral Laurent polynomial $P$ with at most $k$ terms and a $d$-dimensional polytope $\Delta\subset[0,1)^d$, there is a constant $\kappa(d,k)>0$ such that for every torsion point $\omega$ of sufficiently large strictness degree the restricted Galois average differs from the integral over $\Delta$ by $\ll_{\Delta,P}\delta(\omega)^{-\kappa}$. The novelty is that the subset is a polytope rather than the whole cube: the geometry of the polytope enters through a continuous characteristic function and a polytope version of Koksma's inequality, and the singularities of $\log|P|$ near the atoral set are handled with bounded-log truncation and the counting estimates of the full-cube theorem. The paper also computes an explicit exponent and uses it to answer a quantitative height question.

Load-bearing premise

The load-bearing premise is the modulus-of-continuity bound for the continuous characteristic function of the polytope; if the similar-triangle estimate $\rho(\chi^c_{\Delta,\epsilon},t)\le t/(\epsilon\,\mathrm{inrad}(\Delta))$ fails for some polytope, the polytope Koksma inequality loses its power-law error term and Theorem 1.1 collapses.

Editorial extensions

If this is right

For $\Delta=[0,1)^d$, Theorem 1.1 reduces to the quantitative full-cube equidistribution theorem of [DH24], so the polytope statement is a strict generalization.
Since the cube can be partitioned into polytopes, sums over polyhedral pieces of a Galois orbit each converge to the corresponding Lebesgue integral with a power-law error.
Algorithm 1 makes the exponent $\kappa(d,k)$ effectively computable for every dimension and term count, so the rate is not merely existential.
The two-dimensional example gives an explicit quantitative answer to [GS23, Question 6.2]: the height of the intersection point of a line with its torsion translate differs from $2\zeta(3)/(3\zeta(2))$ by $O(\delta(\omega)^{-1/(261\cdot 5^5)})$.

Reading between the lines

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

Because Theorem 3.18 is stated for arbitrary continuous functions with a bounded maximum, the polytope Koksma inequality is portable: any low-discrepancy point sequence, not only Galois orbits, can be fed into it.
The paper notes that its exponent algorithm is likely not optimal, so the true decay in the height application is probably much faster than the stated $1/(261\cdot 5^5)$; optimizing the parameter choices is a natural next step.
The affine construction of $\chi^c_{\Delta,\epsilon}$ only uses the polytope's face structure, so the method should extend to finite unions of polytopes and piecewise-linear boundaries, provided the surface-area estimate in Lemma 3.11 is adjusted.
Replacing essential atorality by explicit sublevel-set volume bounds would give analogues for polynomials whose unit-torus zero set is larger, with exponents depending on those bounds.

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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

The main theorem is the real thing. Section 3 builds a continuous characteristic function for polytopes and proves a polytope version of Koksma's inequality, which is a genuine technical addition to the full-cube result in [DH24]. The modulus-of-continuity bound in Proposition 3.15 is the key step, and it appears to work; the reduction to two-dimensional sections and similar triangles is convincing. Section 4 then packages that inequality with the known discrepancy bounds and the logarithmic-singularity estimates from [DH24] correctly. I agree with the reader that the existence of some positive kappa in Theorem 1.1 is well supported.

The problem is the quantitative "Moreover" and the application in Proposition 1.2. The claimed exponent 1/(261*55) is supposed to come from Algorithm 1, but the algorithm as printed does not produce that number. For d=2, k=2, running the loop literally gives v_2 = 1/512, then v_1 is roughly 1/(512*1600) times a factor of 1/2 from the empty-product convention. The epsilon chosen in line 24 is then dominated by the v_1 term, around 10^-9 or smaller, and so gamma(2,2)=epsilon/16 is at most about 10^-10--not 7*10^-5. If line 24 is read with the typo 'vd 1' as v_d, the answer is still orders of magnitude off. Either lines 21 or 24 have typos, or the numerical constant in Section 5 is wrong. The paper does not give enough information to decide which. This does not affect the existence theorem, but it does invalidate the advertised convergence speed in Proposition 1.2.

A second, smaller issue is that the algorithm's initialization lines (5 and 19) set v_{n-1},...,v_1 to 1 before overwriting them in the while loops; that is probably harmless but adds to the impression that the appendix was not carefully checked.

Who should read this? Number theorists working on equidistribution, heights, or Mahler measures will want the main theorem and the polytope Koksma inequality. The explicit-exponent part should be treated with caution until the algorithm is fixed. I would send this to a serious referee, but with a clear request to check Appendix A and the numerical constant. If the authors can correct the algorithm or revise the claim to a non-explicit kappa, the paper would be solid. As it stands, Proposition 1.2 is not established as stated.

Referee Report

2 major / 4 minor

Summary. The paper proves a quantitative equidistribution theorem for the function log|P| sampled over Galois orbits of torsion points of G_m^d, restricted to the preimage under the cotropicalization map of a d-dimensional polytope Delta contained in [0,1)^d. Theorem 1.1 asserts a power-type convergence rate delta(omega)^{-kappa} for essentially atoral Laurent polynomials P with at most k terms. The proof develops a Koksma-type inequality over polytopes through a continuous characteristic function (Section 3), combines it with a discrepancy bound and the logarithmic-singularity estimates imported from Dimitrov and Habegger (Section 4), and applies the result to a two-dimensional example, yielding Proposition 1.2 on heights of intersections and answering a question of Gualdi and Sombra. Appendix A proposes an algorithm for computing the explicit exponent.

Significance. If the main theorem is correct, it is a substantial quantitative generalization of [DH24, Theorem 1.1], moving from the full cube to arbitrary polytopal subsets, and it provides the first quantitative answer to [GS23, Question 6.2]. The proof strategy is coherent: the polytope Koksma inequality is a natural and useful tool, the discrepancy estimates are cited from the literature, and the reduction to a bounded logarithm near the singular set follows the framework of [DH24]. The paper's value is currently diminished because the advertised explicit exponent, which is the content of Proposition 1.2, is not reproducible from the printed algorithm.

major comments (2)

[Appendix A, Algorithm 1, lines 19-25; Section 5, displayed exponent] The claimed value gamma(2,2) = 1/(261*55) does not follow from Algorithm 1 as printed. For d=2 and k=2, line 19 gives v_2 = 1/(128*4) = 1/512; the while-loop at lines 19-23 then gives v_1 = v_2/(5*1*160) * (1 - 1/2) = 1/(512*1600). In line 24, the first candidate in the minimum is v_1/(2^7*5*4) ~ 4.8*10^{-10} (or, if the garbled expression 'vd 1' is read as v_d, about 7.6*10^{-7}), while the other candidates in the minimum are at least about 6*10^{-5}. Hence epsilon is orders of magnitude smaller than the value epsilon = 16/(261*55) ~ 1.1*10^{-3} that would be needed to obtain gamma(2,2) = 1/(261*55). Since Proposition 1.2 depends on this specific constant, the quantitative form of the Gualdi-Sombra answer is unsupported unless Algorithm 1 is corrected or the value in Section 5 is recomputed. This does not threaten the existence of a positive kappa in Theorem 1.1, but it does affect the advertised explicit convergence speed.
[Section 3.2, Proposition 3.15] The statement and proof of Proposition 3.15 rely on 'sufficiently small' epsilon and t without a quantitative threshold, although the proof of Theorem 3.18 uses the proposition with epsilon = D^{1/(2d+2)} and t = D^{1/(d+1)}. Since the theorem claims an error term for all sufficiently small D, the threshold in Proposition 3.15 should be shown to depend only on the polytope Delta (in particular on inrad(Delta) and diam(Delta)) and not on the particular points x,y in the definition of the modulus of continuity. The reduction step 'when epsilon is small enough we may assume x,y in P_i' is plausible, but a compactness or explicit-geometry argument is needed to make the uniformity clear. This point is load-bearing for the power rate in Theorem 1.1.

minor comments (4)

[Global] There are several typographical errors: 'Laurant' should be 'Laurent' in the Introduction and abstract; 'factes' should be 'facets' in the proof of Theorem 3.18; and 'Vigogradov' should be 'Vinogradov' in Appendix A.
[Section 5] The notation 'Let d = ord(omega)' reuses the fixed dimension d; this should be renamed (for example, n or r) to avoid confusion.
[Appendix A, Algorithm 1] The notation in lines 21 and 24 is hard to parse: the product v_{ell+2}...v_d and the expression 'vd 1' need explicit subscripts and superscripts, and empty products should be defined explicitly.
[Section 3.2] In Proposition 3.15, the phrase 'sufficiently small real numbers epsilon>0 and t>=0' should be replaced by a quantified statement, since the constant in the bound is independent of epsilon and t but the admissible range is not specified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is derived from the external Dimitrov-Habegger full-cube theorem and an independently constructed polytope Koksma inequality, not from its own conclusion.

full rationale

The derivation chain is self-contained in the relevant sense. Theorem 1.1 is proved by combining the polytope Koksma inequality (Theorem 3.18), built on the paper's own continuous characteristic function and modulus-of-continuity estimate (Proposition 3.15), with the full-cube equidistribution theorem of Dimitrov and Habegger imported as Theorem A.1. That external theorem is used as an input, not derived from Theorem 1.1, so the generalization does not presuppose itself. The application in Section 5 uses the limit 2*zeta(3)/(3*zeta(2)) and the non-archimedean height formula from Gualdi and Sombra as external benchmarks, and the three polynomials P1, P2, P3 are checked essentially atoral by the paper's own Example 2.7. The explicit values in Appendix A are a bookkeeping of the Dimitrov-Habegger induction constants, not fitted parameters masquerading as predictions. Remark A.7 candidly states that the algorithm is non-optimal and that a complete description of the convergence speed remains open, which is a limitation, not a circularity. The skeptic's concern that the printed Algorithm 1 does not visibly yield gamma(2,2)=1/(261*55) is an internal reproducibility or typographical issue about a numerical constant; it does not amount to the theorem reducing to its inputs. No load-bearing self-citations or definitional identifications were found.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central theorem rests on external published results from [DH24], [GS23], [KN74], and [BG06] rather than on fitted constants or postulated entities. The continuous characteristic function and truncated hyperpyramids are explicitly constructed and proven, not assumed. The only hand-chosen constants are proof parameters in Algorithm 1 that affect the explicit kappa but not the existence of a positive rate.

free parameters (1)

Auxiliary parameters epsilon and v_1,...,v_d in Algorithm 1
Chosen by hand in Appendix A to satisfy inequalities (17)-(32). They determine the explicit value of gamma(d,k) and hence the numerical exponent in Proposition 1.2, but the existence of some positive kappa in Theorem 1.1 does not depend on their specific values.

assumptions (6)

standard math Full-orbit quantitative equidistribution for essentially atoral P ([DH24, Theorem 8.8 and Theorem A.1])
Used in Section 4 to bound the part of the sum over all Galois conjugates that is not restricted to the polytope.
standard math Discrepancy bound for torsion Galois orbits ([DH24, Proposition 3.3])
Used in Section 4 to convert the discrepancy D into powers of delta(omega).
standard math Counting and volume estimates near atoral sets ([DH24, Lemmas 7.4, 7.5, 7.7, A.3, A.4])
Used in Section 4 to estimate the difference between log and log_r and the measure of the sublevel set where |Q(e(x))| is small.
standard math Isotropic discrepancy bound ([KN74, Theorem 2.1.6])
Used in Section 3 to bound counting errors over convex subsets by a power of the usual discrepancy.
standard math Exact limit and non-archimedean height formulas ([GS23, Proposition 6.1 and Corollary 3.4])
Used in Section 5 to identify the limit 2 zeta(3)/(3 zeta(2)) and the non-archimedean contribution to the height.
standard math Laurent's theorem ensures P(omega^sigma) is nonzero for sufficiently large strictness degree
Used at the start of Section 4 to justify that the summand is well defined for all Galois conjugates.

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Cite this review

Pith. "Pith review of Galois orbits of torsion points over polytopes near atoral sets." pith.science (2026). https://pith.science/paper/SD5MVCFV

@misc{pith2026241211156,
  author       = {Pith},
  title        = {Pith review of: Galois orbits of torsion points over polytopes near atoral sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SD5MVCFV}},
  note         = {Machine review of arXiv:2412.11156}
}

abstract

Given an essentially atoral Laurent polynomial $P$, we show an equidistribution theorem for the function $\operatorname{log}|P|$ on specific subsets of Galois orbits of torsion points of the $d$-dimensional algebraic torus $\mathbb{G}^d_m(\overline{\mathbb{Q}})$. The specific subsets under consideration are the preimages of $d$-dimensional polytopes within the hypercube $[0,1]^d$ under the cotropicalization map. This generalises an equidistribution theorem of V. Dimitrov and P. Habegger, who considered only all Galois orbits that correspond to the entire hypercube $[0,1]^d$. In addition, we provide an estimate for the convergence speed of this equidistribution, expressed as a negative power of the strictness degree. Our approach is to derive an alternative version of Koksma's inequality over polytopes. As an application, we provide the convergence speed of heights on a sequence of projective points for a specific two-dimensional example, answering a question posed by R. Gualdi and M. Sombra. In the appendix, we present an algorithm to compute the explicit value of the power of the strictness degree.

Figures

Figures reproduced from arXiv: 2412.11156 by the authors.

**Figure 2.** An example of the truncated hyperpyramid P ∩ H With these points we have • |x ′ − y ′ | ≤ |x − y| ≤ t, • χ c ∆,ϵ (x ′ ) = χ c ∆,ϵ (x) = 0, χ c ∆,ϵ (y ′ ) = χ c ∆,ϵ (y) = δ ϵ . Set dF B infn |a − b| : a ∈ aff(F ∩ H), b ∈ aff(ϕϵ (F)∩ H) o , DF B infn |a − xc | : a ∈ aff(F ∩ H) o . Then |z − x ′′| = dF. Let x ′′′ be the intersection of F ∩ H and the ray passing through y ′ with start point xc , as in [PITH_FULL_IMAGE:… view at source ↗

**Figure 3.** Similar triangles for computation For the general case x,y ∈ ∆\ri(∆ϵ ), the argument is essentially the same and we still get [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: The partition of [0,1]2 into triangles Furthermore, for any point in the boundaries of these triangles, its image under the complex exponential map [0,1)2 → G 2 m(C), (x,y) 7→ (e i2πx, ei2πy) lies in one of the algebraic subgroups H(1,0),H(0,1),H(1,1),H(1,−1),H(2,−1),H…

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Heights of complete intersections in toric varieties
math.AG 2024-12 accept novelty 5.0 of 10

The height of intersections of two hypersurfaces twisted by torsion points on a toric variety converges to an adelic sum of mixed integrals of roof and Ronkin functions.

Reference graph

Works this paper leans on

18 extracted references · 16 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Brandolini, L

L. Brandolini, L. Colzani, G. Gigante, and G. Travaglini. On the Koksma - Hlawka inequality. J. Complexity , 29(2):158--172, 2013

work page 2013

[2] [2]

Bombieri and W

E. Bombieri and W. Gubler. Heights in Diophantine geometry , volume 4 of New Math. Monogr. Cambridge: Cambridge University Press, 2006

work page 2006

[3] [3]

Brunault, A

F. Brunault, A. Guilloux, M. Mehrabdollahei, and R. Pengo. Limits of Mahler measures in multiple variables. Ann. Inst. Fourier , 74(4):1407--1450, 2024

work page 2024

[4] [4]

Y. Bilu. Limit distribution of small points on algebraic tori. Duke Math. J. , 89(3):465--476, 1997

work page 1997

[5] [5]

Br ndsted

A. Br ndsted. An introduction to convex polytopes , volume 90 of Grad. Texts Math. Springer, Cham, 1983

work page 1983

[6] [6]

J. W. S. Cassels. An introduction to the geometry of numbers. Class. Math. Berlin: Springer, repr. of the 1971 ed. edition, 1997

work page 1971

[7] [7]

J. D. Condon. Asymptotic expansion of the difference of two Mahler measures. J. Number Theory , 132(9):1962--1983, 2012

work page 1962

[8] [8]

Dimitrov and P

V. Dimitrov and P. Habegger. Galois orbits of torsion points near atoral sets. Algebra & Number Theory , 18(11):1945--2001, 2024

work page 1945