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The asymptotic Mahler measure of Gaussian periods

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Pith's one-line read The paper proves that the Mahler measure of Gaussian periods grows asymptotically as n times the multivariate Mahler measure of a cyclovariety, with an explicit error bound.

desk verdict Strong paper with a novel asymptotic identification; the advertised error term leans on an imported constant that isn't re-proved, but the core result holds up. read the letter →

arxiv 2507.09303 v3 pith:T52OE3ND submitted 2025-07-12 math.NT

Gunther Cornelissen , David Hokken , Berend Ringeling This is my paper
classification math.NT MSC 11R0611R1811K3814J3352B2060G50
keywords MahlermeasureGaussianperiodscyclotomicintegersWeilheightlogCalabi-Yauvarietiesalgebraicdynamicalsystemsquantitativeequidistributionrandomwalks
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

Gaussian periods are trace-of-root-of-unity integers that generate cyclic extensions of the rationals; the paper's main theorem states that, for each fixed $k\ge2$, as the conductor grows through values with $\kappa(n)=k$, the logarithmic Mahler measure $m(\alpha_n)$ of the period is asymptotic to $n\,m(C_k)$, where $C_k$ is a specific Laurent hypersurface built from the $k$-th roots of unity. This gives an explicit rate $|m(\alpha_n)/n - m(C_k)|\le 4\sqrt{3k(k+1)}/p^{1/\phi(k)} + 2\log(k)/p$ with $p=kn+1$. The identification turns the old question of minimal height in cyclic Galois extensions into a comparison between a discrete average over residues and an integral against Haar measure on a torus.

What carries the argument

The central object is the cyclovariety $C_k$, the hypersurface $x_0+F_k(x_1,\dots,x_d)=0$ whose Laurent polynomial $F_k$ has exponent vectors given by the coordinates of all $k$-th roots of unity in a basis of $\mathbb{Z}[\zeta_k]$ ($d=\phi(k)$). The Gaussian period $\alpha_n=\sum_{\sigma\in H}\zeta_p^\sigma$ is the trace over the unique subgroup $H\subset(\mathbb{Z}/p)^\times$ of order $k$, so its conjugates are the values of $F_k$ at an equidistributed and structured set of points. The argument carries through the equality of the relevant measures and the transfer of a quantitative equidistribution bound from the finite set of $k$-th roots of unity to those sample points.

What would settle it

Fix $k=4$ and compute $m(\alpha_n)/n$ with high precision for all $n\in N_4$ up to $p\approx10^6$; the claimed inequality predicts that $|m(\alpha_n)/n - m(C_4)| \le 4\sqrt{3\cdot4\cdot5}/p^{1/2} + 2\log4/p$. Any persistent violation above that envelope would disprove the quantitative Theorem B. A more direct test is to recompute the constant $C_Z$ for $Z=\mu_k$ in the cited preprint's normalization in the specific case $k=2$ or $k=4$; if it is not 1, the explicit bound (1.5) needs correction.

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Extended reading notes

Core claim

On the paper's own terms: the logarithmic Mahler measure (equivalently, the Weil height) of the Gaussian period $\alpha_n$ is, up to smaller-order error, exactly $n$ times the multivariate Mahler measure of the cyclovariety $C_k$ when $n$ runs through the set $N_k$ on which the least prime in $kn+1$ is exactly $k$. The proof shows $m(\alpha_n)/n$ is a Riemann sum or a push-forward delta measure of the same one-variable function $\log^+|F_k|$, while $m(C_k)$ is the corresponding integral against the normalized Haar measure; quantitative equidistribution—via Wasserstein distance and a Weyl-type theorem for the sample points—controls the difference.

Load-bearing premise

The quantitative error bound in the main theorem rests on an imported equidistribution estimate for the set of $k$-th roots of unity, including the claim that a certain constant equals 1; if that constant differs, or if the measure identification fails, the explicit decay in $p^{-1/\phi(k)}$ would not hold as stated, though the qualitative limit would survive by the classical equidistribution theorem.

Editorial extensions

If this is right

  • For each fixed $k$, the height of $\alpha_n$ is asymptotic to $n\,m(C_k)$, so the minimal non-cyclotomic height in cyclic degree-$n$ extensions grows linearly in $n$ along every fixed $k$.
  • Combined with the proven asymptotic for $m(C_k)$ when $k=2q^r$ or $k=q$, the leading growth is $(n/2)\log\kappa(n)$ along those subsequences, pinning down lower-order terms as well.
  • Conjecture G asserts that for almost all odd $n$ the true growth is $m(\alpha_n)\asymp n\log\log n$, which would make the linear lower bound in (1.10) tight up to a $\log\log n$ factor for typical odd $n$.
  • The cyclovariety supports a $\mathbb{Z}^{d+1}$-algebraic dynamical system of positive entropy $m(C_k)$, which is Bernoulli and (for a large family of $k$) non-expansive, so the asymptotic constant has a dynamical reading.
  • For $k=2$, the paper's Euler–Maclaurin argument gives the subleading constant $(\log 2)/2$ in the expansion of $m(\alpha_n)$.

Reading between the lines

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

  • If the paper's dictionary is correct, a comparable construction for other prescribed Galois groups should yield hypersurfaces whose Mahler measure, times the degree, is the asymptotic height of an extremal generator; checking this for dihedral or elementary abelian groups is a direct test of the paper's general thesis.
  • The explicit error terms in Theorem B are only as trustworthy as the imported equidistribution constant that the authors assume equals 1; a reader who wants the qualitative result can rely on the classical equidistribution proof, but the claimed uniformity in $k$ and $p$ should be checked against the Hlawka–Koksma bound that the paper derives separately.
  • The random-walk formulation suggests that the asymptotic of $m(C_k)$ for all $k$ follows from a local central-limit statement for the distance of a correlated walk on the cyclopolytope; the Bessel-function representation in Section 6 is a natural starting point for that estimate.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The manuscript studies the asymptotic Mahler measure of the Gaussian periods α_n = tr_H(ζ_p), where p = κ(n)n + 1 is prime and κ(n) is the least k for which kn+1 is prime. For fixed k and n ∈ N_k, it proves that m(α_n) ∼ m(C_k)n, where C_k is the Laurent hypersurface x_0 + F_k(x) = 0 whose Newton polytope is the cyclopolytope, and it gives an explicit k-uniform error bound. The proof compares the empirical measure of the Gaussian periods with the push-forward of Haar measure under F_k. The paper also proves geometric properties of C_k (reflexive polytopes, log Calabi–Yau compactifications), interprets C_k as an algebraic dynamical system with entropy m(C_k), proves asymptotic expansions of m(C_k) for k = 2q^r, k = 2^r and k = q using Bessel functions, develops an algorithm for high-precision computation of m(C_k), proves by computer search that m_q = m(α_q) for q = 3,5,7, and finally derives a conditional framework under which m(α_n) ≍ n log log n for almost all odd n.

Significance. The paper's central achievement is a parameter-free asymptotic identification between the height of a natural family of cyclotomic integers and the multivariate Mahler measure of a family of log Calabi–Yau varieties. The qualitative limit is independently supported by Myerson's equidistribution theorem, and the paper is careful to label its conjectural input. The geometric and dynamical interpretations of the cyclovariety are valuable, and the computer-assisted minimality results in Section 8 are documented with publicly available code. The main caveat is that the explicit quantitative rate in Theorem B, and the constant terms in Theorem E, depend on imported results whose precise hypotheses and normalizations are not verified in the text.

major comments (2)
  1. [§3.2, Eqs. (1.5) and (3.2)] The advertised explicit bound (1.5) is obtained by citing [61, Cor. 3.6 and Remark 3.7(2)] with Z equal to the set of k-th roots of unity. The proof asserts, without re-derivation, that the measure ν_q of [61] is exactly ν_1 = (1/p)∑_{a∈F_p} δ_{tr_H(ζ_p^a)}, that the limit measure μ_g equals F_k^* μ_{T^d}, and that the constant C_Z from [61, Lem. 3.3] equals 1 for this Z. Each of these identifications feeds directly into the right-hand side of (1.5): a different normalization would shift the (log k)/p term, a different limit measure would change m(C_k), and a C_Z different from 1 would rescale the constant 4√(3k(k+1)). Since Theorem B advertises an explicit k-uniform error term, the authors should either prove these identifications in the present paper, state the exact form of the imported result with all hypotheses verified, or weaken the statement to the qualitative limit (1.4), which is independently secured by Myerson's equidistribution and the Hlawka–Koksma argument in §3.3.
  2. [§6.3, proof of Theorem E] The proof of the asymptotic expansion (1.7) reduces to the evaluation ∫_0^∞ log(x) √(2/(πk)) e^{-x^2/(2k)} dx = (1/2)log k − (1/2)log 2 − (1/2)γ, which is stated in the text as 'computed the last integral using Mathematica'. This integral determines the constant terms −γ/2 and −(log 2)/2 in Theorem E, so it should be proved directly (for example, by differentiating the Gamma integral) or a precise reference should be supplied. Similarly, the bounds from [69] used in Lemmas 6.5.1 and 6.5.2 should be quoted with their exact constants and hypotheses, since they control the claimed O(k^{-3/2}) uniformity in x.
minor comments (5)
  1. [§2.1] In the display following (2.1), the inequality '#N_k(X) ⩽ N'_k(X)' should read '#N_k(X) ⩽ #N'_k(X)'.
  2. [§4.3] The sentence 'Since P_k is projective smooth toric by Lemma 4.2.1(i)' is inaccurate for k = 2q^r with r > 1; smoothness in that case follows from Lemma 4.2.1(ii), so the reference should cover both parts.
  3. [References] References [19] and [20] appear to be the same article by Boyd; one of them should be replaced by the intended reference or merged.
  4. [§8.4] The term 'außerwesentliche Diskriminantenteiler' in Lemma 8.4.1 is not standard in English-language papers; a parenthetical translation or definition would improve readability.
  5. [§8.5] In the proof of Lemma 8.5.1, the sentence 'q^q divides the coefficients of h(qx)' is a shorthand for a coefficient-by-coefficient divisibility; stating the resulting valuations v_q(b_j) ≥ q − j directly would be clearer.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem B is a proved equidistribution statement; the main asymptotic is not assumed from the definition of C_k.

full rationale

The central derivation is self-contained. Lemma 3.3.3 establishes the exact identity E_p = m(alpha_n)/n - m(C_k), where E_p is the difference between a Riemann sum over the explicitly defined sample set S_p and the torus integral of f = log+|F_k|. The qualitative limit (1.4) then follows from Myerson's equidistribution theorem (Remark 3.3.4), an external result proved in the paper for completeness, and the Hlawka-Koksma bound (3.9) gives a quantitative alternative. No parameter is fitted to the target quantity, and m(C_k) is not defined in terms of m(alpha_n). The fact that C_k is built from the same k-th roots of unity that form the Gaussian periods is the content of the theorem, not a circular assumption: the equality of the relevant measures is exactly the equidistribution being proved. The Wasserstein bound in (1.5) is imported from the external preprint [61] with explicit identifications of nu_1 and nu_2 and the constant C_Z=1; whether those identifications are correct is a matter of verification of an external result, not circularity, and the qualitative theorem does not depend on that error term. The conjectural part (Conjecture G) is explicitly derived from the labeled Conjectures 9.2.1, 9.2.2, and 9.2.3 and is not presented as a proved consequence. No self-citation chain or fitted-input-as-prediction pattern occurs.

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

The unconditional results rely on standard analytic number theory, toric geometry, and special function estimates, all cited. No parameter is fitted to data. The only imported quantitative estimate with nontrivial content is the Kowalski-Untrau Wasserstein bound. The conjectures in Section 9 are made explicit and appear here only as conditional hypotheses for Conjecture G.

assumptions (6)
  • standard math Quantitative Wasserstein equidistribution theorem of Kowalski and Untrau [61, Cor. 3.6 and Remark 3.7(2)]
    Used in Section 3.2 to prove the explicit error bound (1.5) in Theorem B. The paper imports the result and the assertion C_Z=1 from [61, Lem. 3.3] without proof.
  • standard math Siegel-Walfisz theorem with error term uniform in the modulus as stated in [53, Cor. 5.29]
    Used in Proposition 2.1.2 to count N_k(X) with implied constant independent of k.
  • standard math Landau's explicit bounds for modified Bessel functions [69, main results] with constants b=0.674885 and c=0.7857468704
    Used in Section 6.5 to show non-dominant Bessel terms are exponentially small; this is a load-bearing estimate for the density asymptotics in Theorem 6.3.2.
  • standard math Smyth's lower bound, Boyd's reduction inequality, and Lawton's theorem
    Used in Proposition 6.1.1 to prove m(C_k) is bounded away from zero for all k at least 2.
  • standard math Schmidt-Lind-Ward theorem identifying Mahler measure with topological entropy [73, Thm. 3.1]
    Used in Section 5 to translate Mahler measure into entropy of the algebraic dynamical system, yielding Theorem D.
  • ad hoc to paper Conjecture 9.2.3, the quantitative prime tuplets conjecture for linear forms (Bateman-Horn special case)
    Assumed only in Proposition 9.2.4 to derive the conditional Conjecture G; not used for any unconditional theorem. It is explicitly labeled as a conjecture.
invented entities (1)
  • Cyclovariety C_k and cyclopolytope N_k independent evidence
    purpose: Defined in Definition A via roots of unity; C_k is the limiting hypersurface whose Mahler measure governs m(alpha_n)/n, and its toric geometry controls Theorem C and Theorem D.
    C_k is explicitly constructed, and its Mahler measure matches known values m(C_2)=L'(chi_{-3},-1), m(C_3)=-zeta'(-2)/4, and m(C_4) proven by Brunault; the Picard-Fuchs algorithm gives independent high-precision values, so it is a defined object with independent checks rather than an unconstrained postulate.

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Pith. "Pith review of The asymptotic Mahler measure of Gaussian periods." pith.science (2026). https://pith.science/paper/T52OE3ND

@misc{pith2026250709303,
  author       = {Pith},
  title        = {Pith review of: The asymptotic Mahler measure of Gaussian periods},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T52OE3ND}},
  note         = {Machine review of arXiv:2507.09303}
}
read the original abstract

We construct a sequence of cyclotomic integers (Gaussian periods) of particularly small Mahler measure/height. We study the asymptotics of their Mahler measure as a function of their conductor, to find that the growth rate is the (multivariate) Mahler measure of a family of log Calabi-Yau varieties of increasing dimension. In turn, we study the asymptotics of some of these Mahler measures as the dimension increases, as well as properties of the associated algebraic dynamical system. We describe computational experiments that suggest that these cyclotomic integers realise the smallest non-zero logarithmic Mahler measure in the set of algebraic integers with cyclic Galois group of a given odd order. Finally, we discuss some precise conjectures that imply double logarithmic growth for those Mahler measures as a function of that order. The proofs use ideas from the theory of quantitative equidistribution, reflexive polytopes and toric varieties, the theory of random walks, Bessel functions, class field theory, and Linnik's constant.

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