An Improved Convergence Case for Diophantine Approximations on IFS Fractals

TL;DR

This paper advances the understanding of Diophantine approximations on IFS fractals by providing an improved convergence result in higher dimensions, extending prior work by Pollington and Velani.

Contribution

It presents a novel, improved convergence case for Diophantine approximations on IFS fractals in higher dimensions, building on and enhancing previous results.

Findings

Established an improved convergence result for higher-dimensional IFS fractals.

Extended the analogue of Khintchine's theorem to a broader class of fractals.

Demonstrated the limitations of previous results and provided a stronger convergence criterion.

Abstract

The objective of this paper is to (partially) address the issue of finding an analogue to Khintchine's theorem for IFS Fractals. We study the convergence case for Diophantine approximations, and show an improved result for higher dimensions. This matter has been previously studied by Pollington and Velani in arXiv:math/0401149. Pollington and Velani show a similar result to the one in this paper (a Khinchine convergence case) and we shall show how our result is an improvement in the higher dimensional cases.