21 problems

Let $\mathcal{T}_{\chi}$ and $\mathcal{T}_{\psi}$ be the tiling spaces associated with substitutions $\chi$ and $\psi$, and let $\tilde{\chi}$ and $\tilde{\psi}$ denote the associa…

A straight edge of a tile is a line segment in its boundary that cannot be extended without leaving the boundary. A tile is weakly convex if, for every straight edge, the rest of t…

A substitution tiling is locally finite if, up to rigid motions, it contains only finitely many two-tile configurations. A tiling substitution has a linear expansion determined by…

A substitution system is a tiling system with finitely many tile types and a substitution rule, and Algorithm is the paper's refinement procedure that augments each tile type with…

A substitution system is a tiling system equipped with tile substitutions; it preserves adjacency when, for any two tiles $t,t'$ belonging to distinct supertiles $s,s'$, the tiles…

The chair tiling is obtained as a fixed-point tiling of the plane from a single chair-tile substitution. Consider the graph formed by taking only the vertices where $90$-degree edg…

Let $M$ be the substitution operator associated with one of the substitutions constructed in this work, satisfying conditions 1--3. The essential spectral radius conjecture states…

Continuum-many BD classes conjecture. If

For each integer $n\geq 2$, consider cyclotomic aperiodic substitution tilings (CASTs) with dense tile orientations (DTO), finite local complexity (FLC), the minimal inflation mult…

Let $n$ be an even integer with $n\geq 16$ and $n\equiv 0\pmod{4}$. Let $\zeta_{2n}$ be a primitive $2n$th root of unity, let $\overline{\eta}$ denote complex conjugation, and let…

A substitution tiling has a substitution matrix ${\bm M}$ with integer entries and eigenvalues governing its scaling properties. Nonexistence conjecture. There are no limit-periodi…

Let $n$ be the polygon-order parameter, let the cases be those listed in Table, and let $\eta_{\min}$ and the edge configurations be the quantities specified in Tables--. A rhombic…

Let $n$ be odd with $n\geq5$, and consider a CAST as described in the generalized Penrose CAST conjecture, with its prototiles and substitution rules. A CAST has individual dihedra…

Let $n$ be odd with $n\geq5$, and consider the CASTs described in Theorem. Let $\mu_{n,\lfloor n/2\rfloor}$ be the diagonal-based inflation multiplier used there, and let…

A cyclotomic aperiodic substitution tiling (CAST) is a substitution tiling with the cyclotomic structure considered in the paper. For odd $n$, let $D_n$ denote the dihedral symmetr…

A substitution tiling is a tiling obtained by repeatedly expanding tiles by a fixed factor and subdividing each expanded tile into isometric copies of a finite set of prototiles. A…

Pisot Substitution Conjecture. If $\Phi$ is a 1-dimensional irreducible, unimodular, Pisot substitution, then the $\mathbb{R}$-action on $\Omega_{\Phi}$ has pure discrete spectrum.

Let $\Phi$ be a unimodular Pisot family substitution with expansion $\Lambda$, tiling space $\Omega_{\Phi}$, and essential cohomology $H^1_{ess}(\Omega_{\Phi};\mathbb Z)$. The esse…

Let $\Phi$ be a substitution with tiling space $\Omega_{\Phi}$. Let $H^1_{hyp}(\Omega_{\Phi};\mathbb Z)$ be the hyperbolic part of first cohomology, and let…

Let $\Phi$ be a one-dimensional substitution with unimodular irreducible incidence matrix $M$, meaning that the characteristic polynomial of $M$ is irreducible over $\mathbb Q$, an…

Let $\Phi$ be a substitution with tiling space $\Omega_{\Phi}$, and let $H^1_{hyp}$ be the largest subgroup of $H^1(\Omega_{\Phi};\mathbb Z)$ containing the $\Phi^*$-invariant subg…