Intrinsic Diophantine Approximation for overlapping iterated function systems

TL;DR

This paper extends Khintchine's theorem to limsup sets generated by iterated function systems, providing new insights into intrinsic Diophantine approximation on self-similar sets across arbitrary dimensions.

Contribution

It introduces a novel height function based on periodic representations and establishes metric properties of limsup sets without separation conditions.

Findings

Analogues of Khintchine's theorem for these sets

Detailed metric descriptions of limsup sets

Applicability to arbitrary dimensions without separation conditions

Abstract

In this paper we study a family of limsup sets that are defined using iterated function systems. Our main result is an analogue of Khintchine's theorem for these sets. We then apply this result to the topic of intrinsic Diophantine Approximation on self-similar sets. In particular, we define a new height function for an element of $\mathbb{Q}^d$ contained in a self-similar set in terms of its eventually periodic representations. For limsup sets defined with respect to this height function, we obtain a detailed description of their metric properties. The results of this paper hold in arbitrary dimensions and without any separation conditions on the underlying iterated function system.