17 problems

Let $\beta>1$ be the root of … where $m,n\in\mathbb N$ and $m\geq n+2\geq 3$. Let $\operatorname{Fin}(-\beta)$ denote the set of numbers having finite $(-\beta)$-expansions. Ito–Sa…

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…

Let $n$ be an integer with $n\ge 9$, and consider the base-$3$ expansion of $2^n$. Erdős's ternary conjecture. For every $n\ge 9$, this expansion contains the digit $2$. This class…

Hollander's conjecture. If $L$ is regular, there exists $p$ such that the limit

Let $k$ be the length of the vector $\vec{\mathbf{c}}=(c_1,\ldots,c_k)$, and let the algorithm in the proof of Theorem main be the conversion algorithm for $\vec{\mathbf{c}}$-repre…

Let $\beta(N)=\beta^+(N)\cdot\beta^-(N)$ be the base-phi expansion of the number $N$. Let $w$ be a word of length $m$, and let $R_{\cdot w}$ be the sequence of occurrences of numbe…

Negative-column run-length conjecture. For $i<0$, there are either runs of $1$'s of length $L^{\text{P}}_{-i}$, or runs of $1$'s of length $L^{\text{P}}_{-i}$ directly followed by…

Pell-Lucas intervals conjecture. The $\Lambda^{\text{P}}_n$ are the intervals of constant expansion length of the standard silver-mean representation, and the $\Gamma^{\text{P}}_n$…

A rational base numeration system has base $\frac{p}{q}$, and it is expanding when the base satisfies the expanding property discussed above. The preceding results concern, in part…

Let $\zeta\in\mathbb{Z}_2$ be the $2$-adic number defined in Equation, and let $\ell_\zeta(a)$ denote the length of the zero block beginning at the relevant digit position indexed…

Consider the numeration system of Example, and let $\alpha_1$ and $\alpha_2$ denote the exponents governing the growth of the $2$-adic and $3$-adic valuations $\nu_2(U_i)$ and…

Consider the numeration system of Example, and let $\alpha_1$ denote the exponent governing the growth of the $2$-adic valuations $\nu_2(U_i)$ in Lemma. Conjecture on the valuation…

Representation-language conjecture.

Let $G$ and $H$ be expansions whose corresponding recurrence relations have multiplicatively independent dominant roots. Let $H_G(n)$ and $H_H(n)$ denote the relevant expansion com…

Let $T=(T(n))_{n\ge 0}$ be the Tribonacci numeration system built on the language of words over $\{0,1\}$ avoiding three consecutive ones, and let $A_T$ be the corresponding summat…

For an integer base $k\ge 3$, let $A_k(n)$ be the analogue of $A(n)$ obtained from words and subwords in the language $\{1,\ldots,k-1\}\{0,\ldots,k-1\}^*\cup\{\varepsilon\}$. Base-…

Let $\mathcal{G}_{1}^{(0)}$ denote the set of parameters associated with the finiteness property for two-dimensional shift radix systems, and let $\mathcal{G}_C$ be the explicitly…