Typical self-affine sets with non-empty interior

TL;DR

This paper establishes conditions under which self-affine sets generated by certain invertible matrices and translation vectors almost surely have non-empty interior, advancing understanding of fractal geometry in higher dimensions.

Contribution

It provides new sufficient conditions on matrices ensuring that the resulting self-affine sets typically have non-empty interior.

Findings

Identifies conditions on matrices for non-empty interior in self-affine sets

Shows that these conditions hold for almost all translation vectors

Extends previous results to higher-dimensional cases

Abstract

Let $T_1,\ldots, T_m$ be a family of $d\times d$ invertible real matrices with $\|T_i\| <1/2$ for $1\leq i\leq m$. We provide some sufficient conditions on these matrices such that the self-affine set generated by the iterated function system $\{T_ix+a_i\}_{i=1}^m$ on $\Bbb R^d$ has non-empty interior for almost all $(a_1,\ldots, a_m)\in \Bbb R^{md}$.