A geometric proof of Peck's theorem

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.