{"id":"ec51a082-8801-4f14-9fb6-9098994456d8","arxiv_id":"2412.11156","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Torsion-point Galois orbits equidistribute with a power rate even when restricted to polytope preimages under the cotropicalization map.","lead":"This paper proves a quantitative equidistribution theorem for logarithmic values of a Laurent polynomial over parts of Galois orbits of torsion points that land in a polytope, extending a result of Dimitrov and Habegger. It also gives a convergence speed for heights in a two-dimensional example, answering a question of Gualdi and Sombra.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main Theorem 1.1 appears coherent, but the explicit exponent in Proposition 1.2 is not reproducible from Algorithm 1 as printed; the claimed value 1/(261*55) is unsupported.","rationale":"The reader's weakest_assumption was Proposition 3.15, the modulus-of-continuity estimate for the continuous characteristic function. On inspection, that proposition is actually robust: on each truncated hyperpyramid associated to a facet F, the function chi^c is affine with gradient 1/(epsilon*c), where c is the Euclidean distance from the chosen center x_c to the supporting hyperplane of F, and the containment of the maximal L-infinity ball in the polytope forces c >= inrad(Delta). Hence the claimed Lipschitz bound holds, and the complicated similar-triangle reduction in the paper, while hard to follow, is not a genuine source of failure. The central existence result in Theorem 1.1 therefore appears sound. The real concern is the explicit numerical content: the paper's main novelty beyond DH24 includes an explicit power of delta(omega), computed by Algorithm 1 and used to give the numerical exponent in Proposition 1.2. The printed algorithm has serious notational ambiguities, and a direct reading yields a gamma(2,2) inconsistent with the stated 1/(261*55) by several orders of magnitude. Because the reader already issued a CONDITIONAL verdict on exactly this type of issue, my stress-test does not move the verdict; it sharpens the condition: the explicit constant must be recomputed or the algorithm corrected. If the mismatch is confirmed, Proposition 1.2's exponent should be corrected but the main equidistribution theorem would likely survive.","tokens_in":32008,"tokens_out":29542,"duration_ms":262133,"concrete_test":"Implement Algorithm 1 literally for d=2, k=2 from the printed recurrences, resolving the 'vd 1' and 'v^d' notation in lines 21 and 24 in every plausible way, and compare the output for gamma(2,2) with 1/(261*55). Independently, substitute epsilon=16/(261*55) into the simplified inequalities (17), (21), (24), (26), (28), (29), and (31); if any inequality fails, the claimed exponent is inconsistent with the derivation preceding Algorithm 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The existence part of Theorem 1.1 is well supported by the polytope Koksma inequality of Section 3 and the DH24 estimates packaged in Section 4. The load-bearing weakness is the quantitative 'Moreover' clause and its application in Proposition 1.2: the explicit exponent depends entirely on Algorithm 1, whose printed pseudocode is ambiguous and, taken literally, appears to give a different value. For d=2, k=2, the displayed recurrences in Algorithm 1 give v_2=1/(128*4)=1/512 and, by the while-loop at lines 19-23, v_1 roughly 1/(512*1600) (or still smaller under the 'v^d' reading), so the first term in line 24 is about 10^(-9). The remaining candidates in the min are at most about 6*10^(-5), so one obtains gamma(2,2)=epsilon/16 at most about 10^(-6) and plausibly much smaller, not the claimed 1/(261*55) ~ 7*10^(-5). Alternatively, if line 24's garbled 'vd 1' is read as v_d, the output is still off by orders of magnitude. Proposition 1.2, which answers the Gualdi-Sombra question, relies on this specific power. Thus either the algorithm has typos in lines 21 and 24, or the numerical constant in Section 5 is wrong; the paper does not provide enough information to decide. This does not threaten the existence of a positive kappa in Theorem 1.1, but it does threaten the advertised explicit convergence speed and the accuracy of Proposition 1.2.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":32322,"tokens_out":10090,"duration_ms":88032,"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":[{"comment":"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":"Appendix A, Algorithm 1, lines 19-25; Section 5, displayed exponent"},{"comment":"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.","section":"Section 3.2, Proposition 3.15"}],"minor_comments":[{"comment":"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":"Global"},{"comment":"The notation 'Let d = ord(omega)' reuses the fixed dimension d; this should be renamed (for example, n or r) to avoid confusion.","section":"Section 5"},{"comment":"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":"Appendix A, Algorithm 1"},{"comment":"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.","section":"Section 3.2"}],"recommendation":"major_revision","confidential_remarks":"The existence part of Theorem 1.1 appears sound and valuable, and the polytope Koksma inequality is a useful tool. The only substantive obstacle is the explicit constant: either the algorithm or the numerical value in Proposition 1.2 must be corrected. Once the author supplies a corrected computation and clarifies the uniformity in Proposition 3.15, the paper would likely be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nA 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.\n\nWho 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.","headline":"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.","tokens_in":32895,"tokens_out":3057,"would_cite":false,"duration_ms":28413,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11J83","11G50","14G40","37P30","52B11"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Galois equidistribution","torsion points","strictness degree","cotropicalization","atoral Laurent polynomials","Koksma inequality","Mahler measure","heights in projective space"],"falsifier":"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.","tokens_in":31757,"feed_emoji":"📐","tokens_out":10883,"duration_ms":86061,"temperature":0.7,"pith_summary":"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].","feed_headline":"Torsion-point sums over polytopes converge at a proven rate","feed_subtitle":"Galois averages of log|P| over any polytope match Lebesgue averages up to an explicit power of the strictness degree.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $O(\\delta(\\omega)^{-1/(261\\cdot 5^5)})$."],"supporting_citations":[{"why":"Supplies the full-cube quantitative equidistribution theorem that Theorem 1.1 generalizes, along with the discrepancy estimate, bounded-log lemmas, and the induction scheme used for the exponent.","marker":"[DH24]"},{"why":"Poses the height-convergence question answered by Proposition 1.2 and contributes the triangle partition and the zeta-value limit for the example.","marker":"[GS23]"},{"why":"Provides the qualitative Galois equidistribution theorem that this paper makes quantitative over polytopes.","marker":"[Bil97]"},{"why":"Gives the isotropic-discrepancy comparison between convex sets and box discrepancy used inside the polytope Koksma inequality.","marker":"[KN74]"},{"why":"Supplies basic facts on algebraic subgroups, torsion cosets, and height normalizations used throughout.","marker":"[BG06]"},{"why":"Used to guarantee that $P(\\omega^\\sigma)$ does not vanish for torsion points of sufficiently large strictness degree.","marker":"[Lau84]"}],"fun_headline_variants":["Explicit rate for torsion sums over polytopes","Polytope equidistribution gets power-law speed","Beyond the cube: torsion points, proven rates","Galois averages on polytopes: rate revealed","Torsion-point sums: polytope version converges"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Explicit rate for torsion sums over polytopes","Polytope equidistribution gets power-law speed","Beyond the cube: torsion points, proven rates","Galois averages on polytopes: rate revealed","Torsion-point sums: polytope version converges"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000201,"raw_usage":{"total_tokens":1404,"prompt_tokens":994,"completion_tokens":410,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":610,"completion_tokens_details":{"reasoning_tokens":335}},"tokens_in":610,"tokens_out":410,"duration_ms":4293,"temperature":1.0,"reasoning_tokens":335,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T15:16:12.260158+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}