28 problems

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

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…

Fix $\beta$ in the setting above, and let $\mathfrak{R}_\beta(x)$ denote the set of representation branches associated with a point $x$, viewed as a subset of the sequence space…

Let $\beta\in(\beta_*,2)$, and let $S_\beta$ be the shift on the space $\mathcal{U}_\beta$ of sequences giving unique $\beta$-representations. A shift is intrinsically ergodic when…

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…

Let $d53f$ be the transfer operator associated with the greedy $d6fd$-expansion, and let $N$ be the differentiability order in the estimate … When $a_1=1$, the regularity-threshold…

Let $\beta>1$, $0<a1 eq 1$, and let $d53f$ be the transfer operator in the asymptotic estimate … For $F\in C^\infty([0,1])$, the optimal-exponent conjecture. The estimate above onl…

For integers $q\geq 1$ and $m\geq 2$, let $\beta_{q,m}$ be the parameter defined in the paper, let $T_{\beta_{q,m},\alpha}$ be the associated intermediate beta-transformation, and…

For $q_0>1$, let $U_{q_0,\mathcal{K}(q_0)}$ denote the set of unique double base expansions at the parameter $\mathcal{K}(q_0)$. The preceding theorem shows that this set is counta…

Pell-Lucas discrepancy conjecture. The natural numbers $6,12,20,26,34,\dots$ for which

Generalized Beatty conjecture. The positions at which the $1$'s occur in the column of digit $i$ are given by unions of generalized Beatty sequences.

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

Let $\beta_G\approx 1.46557$ be the unique root in $(1,2)$ of $x^3-x^2-1=0$. For $\beta\in(\beta_G,3/2]$, let ${\pazocal{U}}_\beta$ be the set of points in the fat Sierpinski gaske…

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 $\beta$…

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.…

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.

For a base $\beta>1$, let $T_\beta$ and $T_{-\beta}$ denote the positive- and negative-base transformations, respectively. The measurable-isomorphism conjecture. Among all…

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

Counting beta-expansions conjecture. For every such $\beta$, the displayed formula holds for almost all $x\in I_{\beta}$.

Let $\beta>1$ and write the Rényi $\beta$-expansion of unity as $d_\beta(1)=0.t_1t_2t_3\ldots$. The sequence $(t_i)$ is said to have Hadamard gaps when its nonzero positions have m…

A regular Pisot number is a Pisot number belonging to the regular families approaching the limit points $\phi_r$ and $\psi_r$; for each such number, consider the co-factor obtained…

Let $\beta>1$ be a Pisot number of degree $d+1$, and let $\mathcal{R}_\beta$ denote the family of tiles associated with $\beta$ that covers $\mathbb{R}^d$. The number $\beta$ is a…

Let $\beta$ be a Pisot unit and let the greedy $\beta$-transformation be … on $X=[0,1)$. A tiling conjecture asserts that this transformation produces a tiling for every Pisot unit…