7 problems

Let $\beta$ be a Pisot unit, meaning an algebraic integer greater than $1$ whose other algebraic conjugates have modulus less than $1$ and whose norm is a unit. Call $\beta$ weakly…

Parry-power conjecture. For every $d\geq6$, $4\theta_d4$ is Parry, and for every integer $k\geq2$, $4\theta_d^k4$ is non-Parry.

Non-Parry powers conjecture. For every $d\geq4$ and every integer $k\geq1$, the power $4\beta_d^k4$ is non-Parry.

Let $\beta\in\mathbb{R}_{>1}$, let $c$ be the contraction factor associated with $A_2$, and let ${\cal M}(\theta,c,\beta)$ denote the corresponding switched-system matrix family. I…

Baruchel's conjecture. The digit $d_0(N)=1$ if and only if

Let $1<\beta<\dfrac{1+\sqrt{5}}{2}$, and let $L^{1}(\beta)$ denote the Lebesgue measure of the set of points in $[0,(\beta-1)^{-1}]$ that have a universal expansion in base $\beta$…

Unit-modulus conjugate conjecture. If $\beta$ has a conjugate of modulus $1$, then $W_{\beta,\lceil \beta\rceil-1}$ is not recognizable by a finite automaton.