The Hausdorff dimension of the intersection of \psi-well approximable numbers and self-similar sets

Abstract: Let \psi be a monotonically non-increasing function from N to R, and let \psi_v be defined by \psi_v(q)=1/q^v. Here, we consider self-similar sets whose iterated function systems satisfy the open set condition. For functions \psi that do not decrease too rapidly, we give a conjecturally sharp upper bound on the Hausdorff dimension of the intersection of \psi-well approximable numbers and such self-similar sets. When \psi=\psi_v for some v greater than 1 and sufficiently close to 1, we give a lower bound for this Hausdorff dimension, which asymptotically matches the upper bound as v approaches 1. In particular, we show that the set of very well approximable numbers has full Hausdorff dimension within self-similar sets.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar