Extrinsic Diophantine approximation on manifolds and fractals

Abstract

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 (1984) 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 Jarník–Schmidt and Khinchin theorems from known results.