Construction of numbers with almost all convergents in a Cantor set

TL;DR

This paper constructs numbers within certain Cantor-like sets that are highly well-approximated by rationals inside the set but poorly approximated by rationals outside, with most convergents lying in the set.

Contribution

It generalizes Mahler's question by constructing numbers in missing digit sets with almost all convergents in the set, highlighting new approximation properties.

Findings

Most convergents of the constructed numbers lie in the set

Numbers are arbitrarily well approximable by rationals in the set

Numbers are badly approximable by rationals outside the set

Abstract

In 1984, K. Mahler asked how well elements in the Cantor middle third set can be approximated by rational numbers from that set, and by rational numbers outside of that set. We consider more general missing digit sets $C$ and construct numbers in $C$ that are arbitrarily well approximable by rationals in $C$, but badly approximable by rationals outside of $C$. More precisely, we construct them so that all but finitely many of their convergents lie in $C$.