arXiv:2609.29469v1

A geometric proof of Peck's theorem

math.NT11J1311J6811H06

Paper metadata

Original submission: September 24, 2026
Claimed MSC: 11J13, 11J68, 11H06
arXiv categories: math.NT
Paper license: CC BY 4.0 ↗
Comments: 17 pages, 2 figures

Abstract

Suppose that $d\ge 2$ and that $1,α_1,\ldots ,α_d$ is a basis for a real algebraic number field. A theorem of Peck from 1961 establishes that $\liminf_{n\rightarrow\infty}n\log n\,\|nα_1\|\cdots\|nα_d\|\ < \infty.$ The goal of this paper is to recast Peck's proof in an intuitive geometric framework, where the result follows from a single application of the Minkowski convex body theorem. We will also explain how a simple modification of this approach leads immediately to new proofs of results of de Mathan, Teulié, and Bugeaud regarding the $p$ -adic Littlewood conjecture for $d$ -tuples of algebraic numbers. Finally, changing the shape of the convex body allows us to make progress on a conjecture raised by Peck in the same paper, about distributing the logarithmic savings unequally among the coordinates. We prove the conjecture for every real biquadratic field with its natural basis, and for an arbitrary number field when the factors involved are of comparable size.

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Audited against arXiv v1

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated September 25, 2026

01Statements2 reported findingsNo statement issues found

The geometric form of Peck's theorem, the congruence and p-adic refinements, and the two unequal-weight results all follow with the stated exponents and uniformities. The distinction between bounds at the ambient height $T$ and Peck's literal formulation at the selected denominator $n$ is stated accurately.

Theorems 1 and 2No issue found

The simultaneous-approximation and divisibility conclusions have the stated strength

PDF pages 1-4 and 9-11 · Theorems 1-2 · arXiv:2609.29469v1

The rectified trace lattice turns a balanced unit into an integer denominator and $d$ approximation errors. The unit-lattice volume gives the factor $(\log T)^{-1/(d-1)}$ . Restricting to units congruent to $1$ modulo $m$ makes the trace denominator divisible by $m$ without changing the absolute approximation constant, and the elementary bound on the congruence image of the rank- $r+z$ unit group gives the exponent $1/(r+z)$ in the $p$ -adic estimate.

Exact manuscript, arXiv:2609.29469v1 ↗

Theorems 3 and 5No issue found

The unequal logarithmic savings match the admissible weights

PDF pages 4-5 and 11-16 · Theorems 3 and 5 · arXiv:2609.29469v1

For a general field, the prescribed-side Minkowski box has exactly the required product volume and the comparability assumption controls the quadratic error in exponentiating logarithmic coordinates. In the real biquadratic basis, the two error coordinates depend separately on $y_2$ and $y_3$ through $e^{y_i}-1$ , which removes that quadratic-loss restriction and permits every admissible pair of weights.

02Proofs4 reported findingsNo proof issues found

The lattice, trace-dual, convex-body, congruence-index, and nonlinear-error arguments were checked through their quantitative dependencies. No gap affecting a theorem was found.

Balanced-unit lemmasNo issue found in the reviewed proof

The Minkowski constructions produce nontrivial totally positive units at the claimed height

PDF pages 8-9 and 13-14 · Lemmas 8 and 10 · arXiv:2609.29469v1

The logarithm-and-argument lattices have the asserted rank and covolume. Their symmetric boxes have critical Minkowski volume; the closed-body form of Minkowski's theorem applies (equivalently one may enlarge the box and pass to a limit). The short angle bounds rule out a nonzero lattice point with unit $v=1$ , and squaring makes the resulting unit totally positive while preserving the required height and coordinate bounds.

Trace-lattice conversionNo issue found in the reviewed proof

Balanced conjugates yield the printed approximation errors and a nonzero denominator

PDF pages 6-10 · Lemmas 6-9 and equations (10)-(20) · arXiv:2609.29469v1

The dual-basis identities give $R\sigma(s)=(q,p-q\alpha)$ and isolate the first error direction. Substitution of $s=\alpha_1^*u$ gives the stated error formula. If $q=0$ , every remaining integer coordinate is forced to vanish and $u$ would be a rational integer unit greater than one, a contradiction; hence $n=|q|$ is a valid positive denominator.

Congruence and

p

-adic passageNo issue found in the reviewed proof

The index estimates imply infinitely many denominators and the claimed logarithmic exponent

PDF pages 10-11 · proof of Theorem 2 · arXiv:2609.29469v1

For $u\equiv1\pmod{m\mathbb Z_\Lambda}$ , trace duality forces $m\mid q$ . The congruence subgroup has index at most the size of the finite residue-unit group. For $m=p^k$ , each residue-unit generator has order below $p^{k+d}$ , so the image of the rank- $r+z$ unit group has size $O_p(p^{k(r+z)})$ ; inserting this into the height bound yields $|n|_p\ll(\log n)^{-1/(r+z)}$ . The argument that a fixed denominator cannot persist as $T\to\infty$ correctly supplies infinitely many distinct values.

Unequal-weight estimatesNo issue found in the reviewed proof

The linear and nonlinear error bounds close with the stated hypotheses

PDF pages 11-16 · proofs of Theorems 5 and 3 · arXiv:2609.29469v1

The error map is an isomorphism and its restriction to the trace-zero hyperplane has full rank. In Theorem 5, $f_{\max}\leq c f_{\min}^2$ makes the quadratic exponential remainder no larger than a constant multiple of the target side length. In the biquadratic case, the explicit formulas $\Psi_i(\epsilon)=2\alpha_1^*\alpha_i e^{\zeta_3}(e^{y_i}-1)$ decouple the two saved coordinates, and the constants also make the unsaved auxiliary integer error strictly less than one.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper: arXiv:2609.29469v1
Authors listed: Kavita Dhanda, Josh Flynn, Alan Haynes
Audit date: September 25, 2026

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A geometric proof of Peck's theorem

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The simultaneous-approximation and divisibility conclusions have the stated strength

The unequal logarithmic savings match the admissible weights

The Minkowski constructions produce nontrivial totally positive units at the claimed height

Balanced conjugates yield the printed approximation errors and a nonzero denominator

The index estimates imply infinitely many denominators and the claimed logarithmic exponent

The linear and nonlinear error bounds close with the stated hypotheses

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