Farey nets and multidimensional continued fractions

Abstract

A multidimensional continued fraction algorithm is a generalization of the ordinary continued fraction algorithm which approximates a vector η=(y1,...,yn) by a sequence of vectors\(\left( {\frac{{a_{j,1} }}{{a_{j,n + 1} }}, \ldots ,\frac{{a_{j,n} }}{{a_{j,n + 1} }}} \right)\). If 1,y1,...,yn are linearly independent over the rationals, then we say that the expansion of η isstrongly convergent if $$\mathop {\lim }\limits_{j \to \infty } \left| {\left( {\frac{{a_{j,1} }}{{a_{j,n + 1} }}, \ldots ,\frac{{a_{j,n} }}{{a_{j,n + 1} }}} \right) - \eta } \right| = 0.$$ This means that the algorithm converges at an asymptotically faster rate than would be guaranteed just by picking a denominator at random.