Intrinsic Diophantine approximation by rationals of height with a bounded number of distinct prime factors

TL;DR

This paper investigates the Hausdorff dimension of self-similar sets, including the middle-third Cantor set, that are well-approximated by rationals with bounded prime factors, extending Diophantine approximation theory.

Contribution

It computes the Hausdorff dimension for a broad class of self-similar sets under specific Diophantine approximation conditions involving prime factor restrictions.

Findings

Determined the Hausdorff dimension for rational approximations with bounded prime factors.

Extended previous results to a larger class of self-similar sets.

Provided new insights into the structure of rational approximations within fractal sets.

Abstract

In this article, for a large class of rational self-similar IFS's wich contains the middle-third Cantor set, we compute the Hausdorff dimension of elements a self-similar set that are $\psi$-approximable by rational belonging to this set and satisfying that its numerator has a bounded number of distinct prime divisors. This paper is based on a previous version in which the proof of a lemma (Lemma 3.8) was incorrect.