Local asymptotic Diophantine approximations

The irrationality measure of a real number characterizes its Diophantine approximation property. The problem of counting the number of approximates to a given real number with respect to certain approximation order has been considered by a number of authors such as S. Lang, W. Adams,... We propose a local version of this problem and show that in certain cases the approximates are uniformly distributed, by using standard tools from analytic number theory.