THE STRUCTURE OF HIGHER-DIMENSIONAL NONCOMMUTATIVE TORI AND METRIC DIOPHANTINE APPROXIMATION

We use the cut-down method and metric diophantine approximation techni ques in order to prove that the set of elements Theta = (Theta(jk))(1 less than or equal to j<k less than or equal to N) with the property t hat the noncommutative torus A(Theta) is an inductive limit of direct sums of 2(N-1) circle algebras and the symmetrized noncommutative toru s A(Theta)(sigma) is an AF-algebra is the complement of a first catego ry set in [0, 1)(N(N-1)/2). As a consequence, almost all noncommutativ e tori are classified by their ordered K-theory.