Variations on Dirichlet's theorem

Abstract

We give a necessary and sufficient condition for the following property of an integer d∈N and a pair (a,A)∈R2: There exist κ>0 and Q0∈N such that for all x∈Rd and Q≥Q0, there exists p/q∈Qd such that 1≤q≤Q and ‖x−p/q‖≤κq−aQ−A. This generalizes Dirichlet's theorem, which states that this property holds (with κ=Q0=1) when a=1 and A=1/d. We also analyze the set of exceptions in those cases where the statement does not hold, showing that they form a comeager set. This is also true if Rd is replaced by an appropriate “Diophantine space”, such as a nonsingular rational quadratic hypersurface which contains rational points. Finally, in the case d=1 we describe the set of exceptions in terms of classical Diophantine conditions.