Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6415325 | Journal of Number Theory | 2016 | 12 Pages |
We give a necessary and sufficient condition for the following property of an integer dâN and a pair (a,A)âR2: There exist κ>0 and Q0âN such that for all xâRd and Qâ¥Q0, there exists p/qâQd such that 1â¤qâ¤Q and âxâp/qââ¤ÎºqâaQâA. 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.