Geodesic multidimensional continued fractions

A multidimensional continued fraction expansion is given which finds provably good Diophantine approximations for all 6 e U'1. For any Q> 1 it finds some approximation (p, q) e Z ' with 1 s £ q = £ Q satisfying \\q %- p| | = £ V j + 1 Q~xld. This expansion consists of a sequence of reduced lattice bases for a parametrized series of lattice bases B,(8) (of different lattices) in GL(d + 1,0?), where the positive real parameter / varies. This parametrized family B,(&) forms a geodesic in GL(d + 1,0?), and also projects to a geodesic ge in the Riemannian symmetric space 9>dJrX of all positive definite symmetric matrices. The multidimensional continued fraction expansion is a 'cutting sequence ' expansion for g0 using a Minkowski fundamental domain of GL(d + 1, 2) \ ^ r f + 1. This method generalizes to give continued fraction expansions finding good Diophantine approximations to an arbitrary set of linear forms. 1