6 problems

Let $Q:{\mathbb R}^d\to {\mathbb R}^d$ be an expanding linear map. A pseudo-self-affine tiling with expansion $Q$ is a repetitive FLC tiling ${\mathcal T}$ of ${\mathbb R}^d$ such…

Let $\mathcal T$ be a $d$-dimensional non-polygonal self-affine tiling, and let each tile be $T=(A,i)$. Urbański's boundary dimension conjecture. Each tile satisfies … This folklor…

Sufficiency conjecture. The condition on $\phi$ in the theorem is also sufficient for $\phi$ to be an expansion map of a self-affine tiling of $\mathbb R^d$.

Consider the two matrices … A self-affine tiling (SAT) in $\mathbb{R}^3$ is a tiling with finitely many tile types up to translation and an expansion matrix. Nonexistence conjectur…

Let $phi$ be a linear map on $\mathbb{R}^n$, and let an SAT mean a self-affine tiling with expansion map $phi$. There is an integer matrix $M$ acting on $\mathbb{R}^N$ for some…

Let $\phi$ be a linear expansion map for a self-affine tiling of $\mathbb{R}^n$. The paper establishes a necessary condition on $\phi$: assuming that $\phi$ is diagonalizable over…