Diophantine approximation by almost equilateral triangles

A {it two-dimensional continued fraction expansion} is a map $mu$ assigning to every $x inmathbb R^2setminusmathbb Q^2$ a sequence $mu(x)=T_0,T_1,dots$ of triangles $T_n$ with vertices $x_{ni}=(p_{ni}/d_{ni},q_{ni}/d_{ni})inmathbb Q^2, d_{ni}>0, p_{ni}, q_{ni}, d_{ni}in mathbb Z,$ $i=1,2,3$, 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 $mu^*$ 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 $lim_{nto infty}frac{sum_{i=1}^3dist(x,x_{ni})}{(2d_{n1}d_{n2}d_{n3})^{-1/2}},$ (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.