A class of self-affine tiles in $\mathbb{R}^d$ that are $d$-dimensional tame balls

TL;DR

This paper characterizes when certain self-affine tiles in higher dimensions are topologically equivalent to $d$-dimensional balls, extending previous two- and three-dimensional results using topological and geometric tools.

Contribution

It provides necessary and sufficient conditions for these tiles to be $d$-dimensional tame balls, confirming a conjecture about their homeomorphism to $d$-balls based on interior connectivity.

Findings

Identifies conditions for self-affine tiles to be $d$-dimensional tame balls.

Extends previous results from $ ext{R}^2$ and $ ext{R}^3$ to higher dimensions.

Confirms a conjecture relating interior connectivity to topological type.

Abstract

We study a family of self-affine tiles in $\mathbb{R}^d$ ($d\ge2$) with noncollinear digit sets, which naturally generalizes a class studied originally by Deng and Lau in $\mathbb{R}^2$ and its extension to $\mathbb{R}^3}$ by the authors. By using Brouwer's invariance of domain theorem, along with a tool which we call horizontal distance, we obtain necessary and sufficient conditions for the tiles to be $d$-dimensional tame balls. This answers positively the conjecture in an earlier paper by the authors stating that a member in a certain class of self-affine tiles is homeomorphic to a $d$-dimensional ball if and only if its interior is connected.