Fix d∈N, and let S⊆Rd be either a
real-analytic manifold or the limit set of an iterated function system (for
example, S could be the Cantor set or the von Koch snowflake). An extrinsic
Diophantine approximation to a point x∈S is a rational point
p/q close to x which lies outside of S. These
approximations correspond to a question asked by K. Mahler ('84) regarding the
Cantor set. Our main result is an extrinsic analogue of Dirichlet's theorem.
Specifically, we prove that if S does not contain a line segment, then for
every x∈S∖Qd, there exists C>0 such that
infinitely many vectors p/q∈Qd∖S satisfy
∥x−p/q∥<C/q(d+1)/d. As this formula agrees with
Dirichlet's theorem in Rd up to a multiplicative constant, one
concludes that the set of rational approximants to points in S which lie
outside of S is large. Furthermore, we deduce extrinsic analogues of the
Jarn\'ik--Schmidt and Khinchin theorems from known results