FAREY NETS AND MULTIDIMENSIONAL CONTINUED FRACTIONS

A multidimensional continued fraction algorithm is a generalization of the ordinary continued fraction algorithm which approximates a vector eta = (y1, ..., y(n)) by a sequence of vectors (a(j,1)/a(j,n+1), ..., a(j,n)/a(j,n+1)). If 1, y1, ..., y(n) are linearly independent over the rationals, then we say that the expansion of eta is strongly convergent if [GRAPHICS] This means that the algorithm converges at an asymptotically faster rate than would be guaranteed just by picking a denominator at random. The ordinary continued fraction algorithm can be defined using the Farey sequence, approximating a number by the endpoints of intervals which contain it. Analogously, we can define a Farey net F(n,m) to be a triangulation of the set of all vectors (a1/a(n+1), ..., a(n)/a(n+1)) with a(n+1) less-than-than-or-equal-to m into simplices of determinant +/- 1, and use this algorithm to define a multidimensional continued fraction for eta in which the approximations are the vertices of the simplices containing eta in a sequence of Farey nets. The concept of a Farey net was proposed by A. Hurwitz, and R. Monkemeyer developed a specific continued fraction algorithm based on it. We show that Monkemeyer's algorithm discovers dependencies among the coordinates of eta in two dimensions, but that no continued fraction algorithm based on Farey nets can discover dependencies in three or more dimensions, and none can be strongly convergent, even in two dimensions. Thus there are no good multidimensional algorithms based on Farey nets.