4 problems

Let $\beta>1$ be the root of … where $m,n\in\mathbb N$ and $m\geq n+2\geq 3$. Let $\operatorname{Fin}(-\beta)$ denote the set of numbers having finite $(-\beta)$-expansions. Ito–Sa…

Let $\beta>1$ and suppose that … Here $d_{-\beta}$ denotes the greedy $(-\beta)$-expansion and $0^\omega$ is the infinite repetition of zeros. Purely periodic expansion conjecture.…

Let $\beta$ be a base for which the expansion of $\frac{-\beta}{\beta+1}$ in base $-\beta$ is finite, say … Here ${\rm Fin(-\beta)}$ denotes the set of numbers with finite…

Canonical substitution conjecture. The sequence of distances between consecutive $(-\beta)$-integers can be coded by a fixed point of a canonical substitution that is primitive and…