An analogue of Khintchine's theorem for self-conformal sets

TL;DR

This paper extends Khintchine's theorem to self-conformal sets generated by iterated function systems, establishing conditions for measure-theoretic approximation properties and exploring the existence of exceptionally well-approximated points.

Contribution

It introduces the concept of approximation regularity for IFSs, proves it for conformal IFSs satisfying the open set condition, and formulates an analogue of the Duffin-Schaeffer conjecture within this context.

Findings

IFS with conformal mappings and open set condition are approximation regular.

An analogue of the Duffin-Schaeffer conjecture holds for full Hausdorff dimension.

Existence of points in IFS attractors that are very well approximated, including solutions to Mahler's problem.

Abstract

Khintchine's theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the convergence/divergence of naturally occurring volume sums. In this paper we ask whether an analogous result holds for iterated function systems (IFSs). We say that an IFS is approximation regular if we observe Khintchine type behaviour, i.e., if the size of certain limsup sets defined using the IFS is determined by the convergence/divergence of naturally occurring sums. We prove that an IFS is approximation regular if it consists of conformal mappings and satisfies the open set condition. The divergence condition we introduce incorporates the inhomogeneity present within the IFS. We demonstrate via an example that such an approach is essential. We also formulate an analogue of the Duffin-Schaeffer conjecture and show that it holds for a set of full Hausdorff dimension. Combining our results with the mass transference principle of Beresnevich and Velani \cite{BerVel}, we prove a general result that implies the existence of exceptional points within the attractor of our IFS. These points are exceptional in the sense that they are "very well approximated". As a corollary of this result, we obtain a general solution to a problem of Mahler, and prove that there are badly approximable numbers that are very well approximated by quadratic irrationals. The ideas put forward in this paper are introduced in the general setting of IFSs that may contain overlaps. We believe that by viewing IFS's from the perspective of metric number theory, one can gain a greater insight into the extent to which they overlap. The results of this paper should be interpreted as a first step in this investigation.