A geometric proof of Peck's theorem

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 framewo