Dirichlet is not just bad and singular in many rational IFS fractals

TL;DR

This paper constructs specific vectors within certain rational IFS fractals that are Dirichlet improvable but not singular, extending classical missing digit fractals and exploring their dimensional properties.

Contribution

It introduces new examples of vectors in rational IFS fractals that challenge traditional notions of singularity and Dirichlet improvability, extending known fractal classes.

Findings

Constructed vectors in rational IFS fractals with specific Diophantine properties

Extended results from missing digit fractals to broader rational IFS classes

Discussed lower bounds for Hausdorff and packing dimensions of folklore sets

Abstract

For $m\ge 2$, consider $K$ the $m$-fold Cartesian product of the limit set of an IFS of two affine maps with rational coefficients. If the contraction rates of the IFS are reciprocals of integers, and $K$ does not degenerate to singleton, we construct vectors in $K$ that lie within the ``folklore set'' as defined by Beresnevich et al., meaning they are Dirichlet improvable but not singular or badly approximable (in fact our examples are Liouville vectors). We further address the topic of lower bounds for the Hausdorff and packing dimension of these folklore sets within $K$, however we do not compute bounds explicitly. Our class of fractals extends (Cartesian products of) classical missing digit fractals, for which analogous results had recently been obtained.