  Under a map T, a point x recurs at rate given by a sequence {r_n} near a
point x_0 if d(T^n(x),x_0)< r_n infinitely often. Let us fix x_0, and consider
the set of those x's. In this paper, we study the size of this set for
expanding maps and obtain its measure and sharp lower bounds on its dimension
involving the entropy of T, the local dimension near x_0 and the upper limit of
1/n log 1/r_n. We apply our results in several concrete examples including
subshifts of finite type, Gauss transformation and inner functions.
