Diophantine approximation with nonsingular integral transformations

Abstract

Let Γ be the multiplicative semigroup of all n × n matrices with integral entries and positive determinant. Let 1 ≤ p ≤ n − 1 andWe consider the componentwise action of Γ on V . Let x ∈ V be such that Γx is dense in V . We discuss the effectiveness of the approximation of any target point y ∈ V by the orbit {γx | γ ∈ Γ}, in terms of ||γ||, and prove in particular that for all x in the complement of a specific null set described in terms of a certain Diophantine condition, the exponent of approximation is (n − p)/p; that is, for any ρ < (n − p)/p, ||γx − y|| < ||γ|| −ρ for infinitely many γ.