12 problems

Almost one-to-one coding conjecture. A Pisot automorphism with associated Pisot number $\beta$ admits an almost $1$-to-$1$ coding by $S_\beta$.

Eigenvalue-form conjecture. Every eigenvalue of $M_n$ has the form $\pm\frac{1}{s_{n,j}}$ for all odd $n\geq 5$. If this holds and $g_n(x)$ is irreducible for odd $n\geq 11$, then…

Given a number $\beta\in(1,2)$, let $\nu_{\beta}$ denote the associated Bernoulli convolution and let $\dim_H$ denote Hausdorff dimension. Large-degree Pisot dimension conjecture.…

Let $\nu_{\beta_1}$ denote the Bernoulli convolution associated with $\beta_1$. Suppose that $\beta_1\in(1,2)$, $\beta_1\neq(1+\sqrt{5})/2$, and that $\beta_1$ has a real conjugate…

Dimension-one conjecture. Under these assumptions,

Let … be the Hardy set, where $\|x\|$ denotes the distance from $x$ to the nearest integer. The folklore conjecture. If $(\lambda, \alpha) \in H$, then $\alpha$ is a Pisot–Vijayara…

Let $\beta$ be a Pisot number, meaning an algebraic integer greater than $1$ whose other Galois conjugates have modulus less than $1$, and let $A_\beta$ denote the associated attra…

Let $\lambda$ be a Pisot–Vijayaraghavan (PV) number, let $f$ be a nonzero refinable distribution for dilation by $\lambda$, and suppose its translations lie in … Here…

Let $\alpha$ be a nonzero real algebraic number and let $\zeta$ be real. Exclude pairs of the form … where $L,M,A,B,g>0$ are integers. The quantity…

Let $\zeta$ be a Pisot number and let $\alpha$ be real. The quantities $\underline{\sigma}(\alpha,\zeta)$ and $\overline{\sigma}(\alpha,\zeta)$ denote the lower and upper rates of…

Let $K$ be a real cubic field, and let $\alpha^{(0)},\alpha^{(1)},\alpha^{(2)}$ be its positive $\mathbb{Q}$-basis. Set … Let $\{\varepsilon_n\}_{n=0}^{\infty}$ be the expansion of…

Pisot Substitution Conjecture. If $\Phi$ is a 1-dimensional irreducible, unimodular, Pisot substitution, then the $\mathbb{R}$-action on $\Omega_{\Phi}$ has pure discrete spectrum.