Multidimensional Farey partitions

Abstract

The linear action of SL( n, ℤ +) induces lattice partitions on the ( n − 1)-dimensional simplex † n−1. The notion of Farey partition raises naturally from a matricial interpretation of the arithmetical Farey sequence of order r. Such sequence is unique and, consequently, the Farey partition of order r on A 1 is unique. In higher dimension no generalized Farey partition is unique. Nevertheless in dimension 3 the number of triangles in the various generalized Farey partitions is always the same which fails to be true in dimension n > 3. Concerning Diophantine approximations, it turns out that the vertices of an n-dimensional Farey partition of order r are the radial projections of the lattice points in ℤ + n ∩ [0, r] n whose coordinates are relatively prime. Moreover, we obtain sequences of multidimensional Farey partitions which converge pointwisely.