Intrinsic approximation for fractals defined by rational iterated function systems - Mahler's research suggestion

TL;DR

This paper establishes new results on intrinsic rational approximation within fractals generated by rational iterated function systems, extending Mahler's conjecture and connecting distribution properties of rationals on these fractals.

Contribution

It provides a Dirichlet type theorem, a Khinchin type theorem, and a conjecture related to the distribution of rationals on fractal limit sets.

Findings

Established a Dirichlet type approximation theorem for fractal limit sets.

Proved a Khinchin type theorem linking distribution of rationals on fractals.

Motivated a conjecture on the distribution of rationals in reduced form on the Cantor set.

Abstract

Following K. Mahler's suggestion for further research on intrinsic approximation on the Cantor ternary set, we obtain a Dirichlet type theorem for the limit sets of rational iterated function systems. We further investigate the behavior of these approximation functions under random translations. We connect the information regarding the distribution of rationals on the limit set encoded in the system to the distribution of rationals in reduced form by proving a Khinchin type theorem. Finally, using a result of S. Ramanujan, we prove a theorem motivating a conjecture regarding the distribution of rationals in reduced form on the Cantor ternary set.