Geodesic Multidimensional Continued Fractions.

A multidimensional continued fraction (MCF) associates to a vector fat theta eta = ( theta sub 1 ,..., theta sub d ) of real numbers a sequence "{" font 6 A sup (n) : n >= 0"}" of partial quotient matrices in GL(d + 1, font 5 Z ), and convergent matrices hP sup (n) = font 6 A sup (n) hA sup (n-1) "..." hA sup (0). The rows of font 6 P sup (n) provide approximations to the line in font 5 R sup d+1 through fat 0 and ( theta eta sub 1 ,..., theta sub d , 1). This paper describes a new class of multidimensional continued fractions based on symbolic dynamics in the quotient space GL(d + 1, font 5 Z ) GL(d + 1, RR ) of a particular geodesic depending on fat theta eta in GL(d + 1, RR ). The Minkowski geodesic multidimensional continued fraction uses symbolic dynamics associated to the Minkowski fundamental domain for GL(d + 1, font 5 Z ) GL (d + 1, RR ). Its partial quotients are drawn from a finite set SIGMA sub {d + 1}, so theta is MCF is of additive type, i.e., it is analogous to the ordinary continued fraction expansion with the intermediate convergents included. This expansion yields better Diophantine approximations to fat theta eta eta eta than are known for other MCF's in all dimensions d >= 3. It finds infinitely many Euclidean norm best simultaneous Diophantine approximations for each fixed d , and these determine the Euclidean norm Diophantine approximation constant. Furthermore for any given B theta eta eta eta eta e first row of some font 6 P sup (n) is an approximation ( p sub 1 ,..., p sub d , q) such theta at 1 = q = B and sum from i=1 to d (q theta eta sub i - p sub i ) sup 2 = (d + 1) B sup {- 2 over d} . An algorithm is described for computing the Minkowski geodesic MCF expansion. This geodesic approach to MCF's is extended to produce continued fraction expansions finding Diophantine approximations to an arbitrary set of linear forms.