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

Let ψ :N→ R₊ be a monotonically non-increasing function, and let ψ ᵥ:N→ R₊ be defined by ψ ᵥ(q)=1/qᵛ. In this article, we consider self-similar sets whose iterated function systems satisfy the open set condition. For functions ψ that do not decrease too rapidly, we give a conjecturally sharp upper bound on the Hausdorff dimension of the intersection of ψ-well approximable numbers and such self-similar sets. When ψ =ψ 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↓ 1. In particular, we show that the set of very well approximable numbers has full Hausdorff dimension within self-similar sets, thus confirming a conjecture of Levesley, Salp, and Velani.