14 problems

Boyd's conjecture. There exist Salem numbers of degree greater than $4$ that are not Parry numbers.

Let $\theta_n$, for $n\geq 2$, be the unique root in $(0,1)$ of $-1+x+x^n$, and call a real algebraic integer greater than $1$ a Salem number when it has the usual Salem-number pro…

Let an arithmetic hyperbolic orbifold of the first type be a quotient of hyperbolic space by an arithmetic lattice of the first type. For a hyperbolic element $\gamma$, write…

Boyd's closure conjecture. The set $4\mathcal{H}_\infty=\mathcal{S}\cup\mathcal{T}4$ is closed.

Strong Lehmer conjecture. The optimal lower-bound constant is $c=\tau_{10}$.

Let $W(E_{10})$ be the Coxeter group under consideration, and let $\rho(d4e)$ denote the spectral radius of an element $d4e\in W(E_{10})$. For each level $n$, let…

Short Geodesic Conjecture. There exists a neighborhood $U\subset\operatorname{SL}_2(\mathbb R)$ of the identity such that $\Gamma\cap U=\{e\}$ for every arithmetic torsion-free coc…

Let $M(\alpha)$ denote the Mahler measure of an algebraic number, and let a Salem number be a real algebraic integer greater than $1$ whose conjugates lie in the closed unit disk,…

Complex Salem conjecture. There exists $\epsilon>0$ such that every complex Salem number $\alpha$ satisfies

Sury's geometric conjecture. There exists a neighborhood $U\subset\operatorname{SL}_2(\mathbb{R})$ of the identity such that, for every torsion-free cocompact lattice…

Let $\theta_n^{-1}$ denote the relevant Perron-number sequence, and let a Salem number be a real algebraic integer greater than $1$ whose conjugates satisfy the usual Salem-number…

Let ${\rm T}$ be the set of Salem numbers, let $c>1$, and let $A_q$ denote the classes of Salem numbers used in the source. Local Density Conjecture. For every $c>1$, there exists…

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

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…