Metric Diophantine approximation on fractals

Inspired by a problem proposed by Mahler, we will address the following related question, ‘How well can irrationals in a missing digit set be approximated by rationals with polynomial denominators?’ and prove some related results. To achieve this, we will be closely looking at Khintchine’s theorem, particularly the convergence case and aim to prove a Khintchine-like convergence theorem for missing digit sets with large bases and rationals with polynomial denominators.

💡 Research Summary

The paper investigates metric Diophantine approximation on a class of fractal sets known as missing‑digit sets, focusing on approximations by rational numbers whose denominators are values of a fixed polynomial P(q). Inspired by Mahler’s question about approximating points in the middle‑third Cantor set, the author seeks a Khintchine‑type convergence theorem for these fractals when the base b is large and only one digit is omitted.

After recalling the classical Khintchine theorem (convergence case) and the Borel–Cantelli lemma, the author defines the missing‑digit set K_{b,D} as the set of numbers in