Intrinsic Diophantine approximation: a solution to Mahler's problem
Edouard Daviaud
TL;DR
The paper tackles intrinsic Diophantine approximation on self-similar fractals, addressing Mahler's question by computing the Hausdorff dimension of sets of points in a rational self-similar attractor $K_T$ that are $\psi$-approximable by rationals within the same fractal. The authors blend geometric measure theory, discrepancy/equidistribution methods, and fractal separation conditions (AWSC/WSC) to derive sharp dimension formulas for both intrinsic and (where appropriate) extrinsic approximations, including a complete intrinsic-dimension formula $\dim_H E_{\psi,T,\text{int}} = \frac{\dim_H K_T}{\max\{1,\delta_{\psi}\}}$ with $\delta_{\psi}=\liminf_{q\to\infty} \frac{-\log \psi(q)}{\log q}$. A key innovation is the use of AWSC to obtain upper bounds and a mass transference principle to obtain matching lower bounds, without requiring strong separation. The results substantially extend Mahler-type intrinsic approximation to a broad class of rational IFSs and illuminate how intrinsic versus extrinsic approximation scales differ on fractals, with connections to Ahlfors regularity and equidistribution theory.
Abstract
In this article, we compute the Hausdorff dimension of elements of a rational self-similar set that are $ψ$-approximable by rational belonging to the set, this for a large class of self-similar sets, containing in particular the middle-third Cantor set.