Diophantine approximation by almost equilateral triangles

Abstract

A {\it two-dimensional continued fraction expansion} is a map $\mu$ assigning to every $x\in {\mathbb{R}}^2\setminus {\mathbb{Q}}^2$ a sequence $\mu (x)=T_0,T_1,\dots$ of triangles $T_n$ with vertices $x_{ni}=(p_{ni}/d_{ni},q_{ni}/d_{ni})\in {\mathbb{Q}}^2,d_{ni}>0,p_{ni},q_{ni},d_{ni}\in \mathbb{Z},$ $i=1,2,3$, such that \begin{eqnarray*} \det \left(\begin{matrix} p_{n1}& q_{n1} &d_{n1}\\ p_{n2}& q_{n2} &d_{n2}\\ p_{n3}& q_{n3} &d_{n3} \end{matrix} \right) = \pm 1\,\,\, \,\,\,\mbox{and}\,\,\,\,\,\, \bigcap_n T_n = \{x\}. \end{eqnarray*} We construct a two-dimensional continued fraction expansion ${\mu }^{\ast }$ such that for densely many (Turing computable) points $x$ the vertices of the triangles of $\mu (x)$ strongly converge to $x$. Strong convergence depends on the value of $\lim_{n\to \infty}\frac{\sum_{i=1}^3\dist(x,x_{ni})}{(2d_{n1}d_{n2}d_{n3})^{-1/2}},$ ("dist" denoting euclidean distance) which in turn depends on the smallest angle of $T_n$. Our proofs combine a classical theorem of Davenport Mahler in diophantine approximation, with the algorithmic resolution of toric singularities in the equivalent framework of regular fans and their stellar operations.