Author

Alan Haynes

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Preprint Oct 2026

Gaps theorems for linear forms with cubic coefficients

Let $1,\alpha_1,\alpha_2$ be a basis for a cubic number field $K\subset\mathbb{R}$, and for $\tau_1,\tau_2\ge 1$ let $G(\mathbf{\alpha},\mathbf{\tau})$ denote the number of distinct gaps between the fractional parts of the numbers $m_1\alpha_1+m_2\alpha_2$, with $m_1,m_2\in\mathbb{Z}$, $0\le m_1<\tau_1$ and $0\le m_2<\...

Alan Haynes · 0 citations

Preprint Sep 2026

A geometric proof of Peck's theorem

Suppose that $d\ge 2$ and that $1,\alpha_1,\ldots ,\alpha_d$ is a basis for a real algebraic number field. A theorem of Peck from 1961 establishes that \[ \liminf_{n\rightarrow\infty}n\log n\,\|n\alpha_1\|\cdots\|n\alpha_d\|\<\infty.\] The goal of this paper is to recast Peck's proof in an intuitive geometric framework...

Kavita Dhanda, J. Flynn, Alan Haynes · 1 citation · ⚡1