11 problems

SRS weak-tiling conjecture. The collection

For real numbers $S_0,S_1$ and initial values $s_0,s_1\in\mathbb{Z}$ satisfying … let $(s_n)$ be the resulting sequence of integers. In the case $d=2$, $S_0=1$, and $|S_1|<2$, peri…

Let $\mathcal{G}_{1}^{(0)}$ denote the set of parameters associated with the finiteness property for two-dimensional shift radix systems, and let $\mathcal{G}_C$ be the explicitly…

SRS tile boundary conjecture. For every $\mathbf{x}\in\mathbb{Z}^d$,

Akiyama's finiteness-property conjecture. For every $d\in\mathbb{N}$,

SRS tile tiling conjecture. SRS tiles always induce tilings of their representation spaces.

Bertrand–Schmidt conjecture. Each element of $\mathbb{Q}(\beta)$ admits a periodic beta-expansion.

Let $\mathbf{r}\in\mathcal{E}_n=\{\mathbf{r}\in\mathbb{R}^n:\varrho(\mathbf{M}_\mathbf{r})<1\}$, where $\mathbf{M}_\mathbf{r}$ is the companion matrix of $\mathbf{r}$, and for…

Let $\mathbf{r}=(r_0,\ldots,r_{d-1})\in\operatorname{int}(\mathcal{D}_d)$ with $r_0\ne0$. Weak-tiling conjecture. The collection … is a weak tiling, meaning a weak $1$-tiling: ever…

Let $\mathbf{r}=(r_0,\ldots,r_{d-1})\in\operatorname{int}(\mathcal{D}_d)$ be a reduced parameter, and let $\mathcal{T}_{\mathbf{r}}(\mathbf{x})$ be the associated SRS tile for…

An SRS tile $\mathcal{T}_{\mathbf{r}}(\mathbf{x})$ is the tile associated with a parameter $\mathbf{r}$ and a vector $\mathbf{x}\in\mathbb{Z}^d$. SRS tile regularity conjecture. An…