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 $\mu^*$ 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.
Categories: math.NT