arXiv:2609.29360v1

Refinements of Peck's theorem on simultaneous approximation to algebraic numbers

math.NT11J1311J6111J68

Paper metadata

Original submission: September 24, 2026
Claimed MSC: 11J13, 11J61, 11J68
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 18 pages

Abstract

Let $n$ be an integer with $n\ge2$ , and let $E$ be a real algebraic number field of degree $n+1$ over ${\mathbb Q}$ . Let $α_1, \ldots , α_n$ be real numbers in $E$ such that $(1,α_1,\ldots,α_n)$ is a linear basis of $E$ over ${\mathbb Q}$ . Let $ν_1, \ldots, ν_{n-1}$ be real numbers satisfying $0<\max_{1\le i\le n-1}ν_i\le2\min_{1\le i\le n-1}ν_i,\quad ν_1+ \ldots +ν_{n-1}=1.$ We establish that there exist a real number $C$ , depending only on $α_1, \ldots , α_n$ , and infinitely many integers $Q \ge 2$ satisfying the inequalities $Q^{1/n}\Vert Qα_i\Vert \le C (\log Q)^{-ν_i}, \quad 1\le i\le n-1, \quad Q^{1/n}\Vert Qα_n\Vert\le C.$ This answers partially a conjecture of Peck, who proved in 1961 this statement in the particular case where $ν_1 = \ldots = ν_{n-1} = 1/ (n-1)$ . We also improve a result from 2004 of de Mathan and Teulié on a question of simultaneous Diophantine approximation involving a non-Archimedean valuation.

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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 26, 2026

01Statements2 reported findingsNo statement issues found

The weighted refinement of Peck's theorem, the improved $p$ -adic logarithmic exponent, and the complementary $p$ -adic lower bound all follow with the stated dependencies. The quantitative hypotheses translate correctly into the exponent condition $\max\nu_i\leq2\min\nu_i$ .

Theorem 1.1No issue found

The unequal-weight simultaneous approximation bounds verify

PDF pages 2 and 4-10 · Theorem 1.1 and Section 2 · arXiv:2609.29360v1

The trace-dual construction makes the first $n-1$ errors linear in the deviations of conjugate ratios. A mixed real-and-angular pigeonhole argument produces a nontrivial unit with linear errors at scales $\varepsilon_i$ and height $O((\varepsilon_1\cdots\varepsilon_{n-1})^{-1})$ . The quadratic remainder is absorbed exactly by $\max\varepsilon_i^2\leq\min\varepsilon_i$ , and Baker's lower bound prevents the selected unit and denominator from remaining bounded.

Exact manuscript, arXiv:2609.29360v1 ↗

Theorems 1.2 and 1.3No issue found

The non-Archimedean upper and lower refinements have the stated exponents

PDF pages 3-4 and 10-17 · Theorems 1.2-1.3 and Sections 3-4 · arXiv:2609.29360v1

The simultaneous archimedean and $p$ -adic pigeonhole step uses $r-1$ real cells and one $p$ -adic cell, giving unit exponents $O(p^{rs})$ . Raising the unit to $p^s$ yields $|Q_0|_p\ll p^{-2s}$ while $\log Q_0\ll p^{(r+1)s}$ , hence the factor $(\log Q)^{2/(r+1)}$ . Conversely, the finite reduction to balanced units and Yu's lower bound for a nonzero $p$ -adic logarithmic form give the claimed negative power of $\log Q_0$ .

02Proofs3 reported findingsNo proof issues found

The dual-basis linear algebra, archimedean and mixed $p$ -adic pigeonhole counts, second-order logarithm estimates, height comparisons, and applications of the cited Baker--Yu bounds form a complete proof chain. No gap affecting a theorem was found.

Archimedean constructionNo issue found in the reviewed proof

The unit selection and second-order error control close

PDF pages 4-10 · Lemmas 2.1-2.2 and completion of Theorem 1.1 · arXiv:2609.29360v1

The conjugate-difference matrix is inverted explicitly using the trace-dual basis. The pigeonhole count has exactly $n$ integer parameters against $n-1$ real constraints, and the proof correctly rules out a solution supported only in the angular variables. Expanding $e^{Y_j}-1=Y_j+O(\varepsilon_1^2)$ then gives the prescribed individual errors. The regulator minor and a nonzero logarithmic form supply both lower and upper height bounds.

Mixed archimedean/

p

-adic constructionNo issue found in the reviewed proof

The cell count and $p^s$ power produce the exponent $2/(r+1)$

PDF pages 10-14 · Lemma 3.1 and proof of Theorem 1.2 · arXiv:2609.29360v1

There are $r-1$ independent real logarithmic-modulus constraints and one rational $p$ -adic constraint. Partitioning each at precision $p^{-s}$ gives $O(M^{r-1}p^{rs})$ cells, so a nonzero exponent vector exists with $M=O(p^{rs})$ . The exact first-order $p$ -adic cancellation improves $Q_0$ from $p^{-s}$ to $p^{-2s}$ ; the treatment of $p=2$ is covered by the strict logarithm convergence radius and the $|2|_p$ factor.

Lower-bound argumentNo issue found in the reviewed proof

The finite-unit reduction and logarithmic-form estimate apply

PDF pages 14-17 · Lemmas 4.1-4.3 and proof of Theorem 1.3 · arXiv:2609.29360v1

The approximation inequalities bound every nonidentity conjugate of the trace-dual element. Division by the balanced unit of matching height leaves an algebraic integer in a fixed compact Minkowski region, hence one of finitely many coefficients $\gamma$ . Lemma 4.3 converts the first $n$ traces into a nonzero $p$ -adic conjugate difference, to which Yu's theorem applies with coefficient height $O(\log|q_0|)$ .

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper: arXiv:2609.29360v1
Authors listed: Yann Bugeaud, Bernard de Mathan
Audit date: September 26, 2026

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Refinements of Peck's theorem on simultaneous approximation to algebraic numbers

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The unequal-weight simultaneous approximation bounds verify

The non-Archimedean upper and lower refinements have the stated exponents

The unit selection and second-order error control close

The cell count and psp^sps power produce the exponent 2/(r+1)2/(r+1)2/(r+1)

The finite-unit reduction and logarithmic-form estimate apply

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The cell count and $p^s$ power produce the exponent $2/(r+1)$