A geometric proof of Peck's theorem
Abstract
Suppose that d≥ 2 and that 1,α1,… ,αd is a basis for a real algebraic number field. A theorem of Peck from 1961 establishes that liminfn→∞nlog n |nα1|⋯|nαd| < ∞. 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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BibTeX
@misc{dhanda2026geometric,
title = {{A geometric proof of Peck's theorem}},
author = {Kavita Dhanda and Josh Flynn and Alan Haynes},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29469},
eprint = {2609.29469},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29469},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A geometric proof of Peck's theorem
AU - Kavita Dhanda
AU - Josh Flynn
AU - Alan Haynes
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29469
UR - https://arxiv.org/abs/2609.29469
AB - Suppose that d≥ 2 and that 1,α_1,… ,α_d is a basis for a real algebraic number field. A theorem of Peck from 1961 establishes that liminf_n→∞nlog n |nα_1|⋯|nα_d| < ∞. 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.
ER -