Alan Haynes

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math.NT2026

Gaps theorems for linear forms with cubic coefficients

Alan Haynes

Let $1, α_{1}, α_{2}$ be a basis for a cubic number field $K \subset R$ , and for $τ_{1}, τ_{2} \geq 1$ let $G (α, τ)$ denote the number of distinct gaps between the fractiona…

math.NT2026

A geometric proof of Peck's theorem

Kavita Dhanda, Josh Flynn, Alan Haynes

Suppose that $d \geq 2$ and that $1, α_{1}, \dots, α_{d}$ is a basis for a real algebraic number field. A theorem of Peck from 1961 establishes that \[ \liminf_{n\rightarrow\infty}n\log n…

math.NT2023

Accumulation points of normalized approximations

Kavita Dhanda, Alan Haynes

Building on classical aspects of the theory of Diophantine approximation, we consider the collection of all accumulation points of normalized integer vector translates of points $q…