Peck定理的一个几何证明
A geometric proof of Peck's theorem
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中文总结 AI 辅助
本文用Minkowski凸体定理给出Peck定理的几何证明，并推广至p-adic情形及Peck关于对数节省不均匀分配的猜想。
中文摘要 AI 辅助
假设 $d\ge 2$ 且 $1,\alpha_1,\ldots,\alpha_d$ 是实数代数数域的一组基。Peck 在1961年的一则定理确立了 \\[ \liminf_{n\rightarrow\infty}n\log n\\,\\|n\alpha_1\\|\cdots\\|n\alpha_d\\|\\ < \infty.\\] 本文的目标是将 Peck 的证明重新表述在一个直观的几何框架中，使得该结果仅通过一次应用 Minkowski 凸体定理即可得出。我们还将说明，对这一方法进行简单修改即可直接得到 de Mathan、Teulié 和 Bugeaud 关于 $d$ 元代数数的 $p$-adic Littlewood 猜想的结果的新证明。最后，改变凸体的形状使我们能够在 Peck 在同一篇论文中提出的一个猜想上取得进展，该猜想涉及在对数节省在坐标间的不均匀分配。我们证明了该猜想对每个具有自然基的实双二次域成立，并且对任意数域在涉及的因子大小相当的情况下也成立。
英文摘要
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.
