33 problems

Let an inflation tiling be one-dimensional, with inflation factor a Pisot–Vijayaraghavan number and inflation matrix $M$ having an irreducible characteristic polynomial. Here, a Pi…

Lower-bound conjecture. We conjecture that the following bounds hold: for $q\in(1,2)$,

Pisot characterization conjecture. One has $l_n(q)>0$ if and only if $q$ is a Pisot number.

Let $\beta$ be a Pisot number with Property (F), and let $\alpha$ be a conjugate of $\beta$. For $x\in\mathbb{R}$, let $\phantom{}_{\alpha}\#(x)$ denote the number of different…

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

Let $\theta>1$ be an algebraic number that is not a Pisot number, and let $\ell(\theta)$ denote the quantity defined earlier in the paper. Algebraic non-Pisot conjecture. For every…

Let $T$ be a Pisot automorphism of the torus, algebraically conjugate to the companion toral automorphism $T_{\beta}$ of a Pisot unit $b2$, so that there exists…

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

Let $1<\beta<2$ be a Pisot number other than the golden ratio, and let $\mu_{\mathbf{p}}$ be the self-similar measure considered in the paper. Non-doubling conjecture. The measure…

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…

Boyd’s conjecture. The optimal constant $c$ should equal

Let $D_2(\mu_\lambda)$ be the correlation dimension of the Bernoulli convolution measure $\mu_\lambda$, and let $\lambda$ be either the reciprocal of the Fibonacci parameter … or t…

Let $\mathcal A$ be an integer alphabet, let $X_{\mathcal A}(\beta)$ denote the associated spectrum, and consider Pisot numbers $\beta_n$ of increasing degree in an interval…

Let $\epsilon>0$, and let $I_{\beta_n}$ denote the interval associated with the Pisot number $\beta_n$, $\mu_{I_{\beta_n}}$ the corresponding measure, and…

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…

Let $q$ be a Pisot number and let $q'$ denote a conjugate of $q$. Consider the conjectured lower bounds for conjugates in the intervals $(1,2)$, $(2,3)$, and $(m,m+1)$ for $m\geq3$…

Let $m\geq 3$, let $q\in(m,m+1)$ be a Pisot number, and let $q'$ be any conjugate of $q$. Conjectured lower bound. We conjecture that … This extends the proposed uniform lower-boun…

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

A multinacci number is a Pisot number that is the real root greater than $1$ of $\beta^n=\beta^{n-1}+\cdots+\beta+1$ for some integer $n\geq 2$. For such a $\beta$, let $T_\alpha$…

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 $\sigma$ be an irreducible unimodular Pisot substitution. For any two symbols $i,j$ in its alphabet, the strong coincidence condition on prefixes (respectively, on suffixes) me…

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…