A class of self-affine tiles in [formula omitted] that are tame balls revisited.

The author of this paper and coauthors in 2022 studied a family of self-affine tiles in R d with noncollinear digit sets, and gave a sufficient and necessary condition for such tiles to be tame balls. We in this paper mainly present a simpler proof of such equivalent condition. We replace quadric surfaces by some zigzag planes, and redefine the quasi-invariant plane which plays a key role in the construction of the desired homeomorphism. This adjustment greatly simplifies the proof. [ABSTRACT FROM AUTHOR]