The Hausdorff dimension of the intersection of $ψ$-well approximable numbers and self-similar sets
Suxuan Chen
TL;DR
This work studies the intersection of $\\psi$-well approximable numbers with self-similar sets under the open set condition, proving an upper bound for $\\dim_H(W(\\psi)\\cap K)$ when $\\psi$ decays moderately and a near-sharp lower bound for $\\dim_H(W(v)\\cap K)$ as $v\\to1^+$. Leveraging effective equidistribution results for self-similar measures on $\\text{SL}_2(\\mathbb{R})/\\text{SL}_2(\\mathbb{Z})$, a Mass Transference–type approach, and a careful limsup-set construction, the authors translate Diophantine approximation properties into Hausdorff-dimension estimates on fractal sets. A key corollary is that $\\dim_H(\\text{VWA}\\cap K)=\\dim_H(K)$, confirming the Levesley–Salp–Velani conjecture for Cantor-type self-similar sets and extending Khintchine/Jarnik–Besicovitch-type phenomena to fractal geometries. The results bridge Diophantine approximation and fractal geometry via dynamical methods, offering explicit bounds depending on the IFS and yielding sharp dimension statements for both upper and lower bounds in the nontrivial regime.
Abstract
Let $ψ:\mathbb{N}\rightarrow\mathbb{R}_+$ be a monotonically non-increasing function, and let $ψ_v:\mathbb{N}\rightarrow\mathbb{R}_+$ be defined by $ψ_v(q)=1/q^v$. 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 $v\downarrow 1$. 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.