14 problems

Let $f$ be the infinite Fibonacci word, let $C_f$ denote its closed-rich constant, and let $\phi=\frac{1+\sqrt{5}}{2}$ be the golden ratio. Fibonacci closed-rich constant conjectur…

BDL representation conjecture. Such a geometric representation with BDL property exists if and only if at least one eigenvalue of $M_{\psi}$ has modulus strictly smaller than $1$.

Puzynina–Zamboni conjecture. If the palindromic lengths of all factors of $d4ce$ are bounded, then $d4ce$ is ultimately periodic.

Let an infinite word over the alphabet $\{0,1,2\}$ define a hereditary bipartite graph class by taking the finite induced subgraphs of the associated infinite graph whose vertices…

Frid's conjecture. The palindromic length sequence of an infinite word is bounded if and only if the infinite word is ultimately periodic.

Let $u$ be an infinite word that is $p$-power-free for some $p$, meaning that $u$ contains no factor equal to the $p$th power of a nonempty word. Let $PPL_u(n)$ be the minimum numb…

Let $u=u[1]u[2]\cdots$ be an infinite word. For $n\in\mathbb N$, let $PPL_u(n)$ denote the minimum number of concatenated palindromes needed to express the prefix of $u$ of length…

Let $x$ be an infinite word over an alphabet $\mathcal{A}$. A super-monochromatic factorisation of a suffix of $x$ is a factorisation into finite words such that all factors and ev…

Let $\alpha$ be an infinite word over the alphabet $\\{0,1,2\\}$. An infinite word is almost periodic if every factor $\beta$ of $\alpha$ occurs in every sufficiently long factor o…

Fix glides $u$ and $v$, and choose positive parameters for the state ${\bf w}$, while leaving the parameters of ${\bf z}$ unspecified. Let $F$ denote the interaction map. Uniquenes…

Let $v'$ be a primitive glide if there is no glide $v$ with offset $k$ and integer $\ell\geq1$ such that $v'=v\rho^{-k}(v)\dotsc\rho^{-\ell k}(v)$. Let…

Let $n$ be chosen, let $u$ and $v$ be glides with fixed reduced words $u=s_{i_1}\cdots s_{i_l}$ and $v=s_{j_1}\cdots s_{j_m}$, and assume that…

Let $\mathbb{A}$ be a finite alphabet, let $\mathbb{A}^+$ be the set of non-empty finite words over $\mathbb{A}$, and let $\boldsymbol{P}$ be the class of infinite words…

Let $A$ be a finite non-empty set and let $x\in A^\omega$. Write $\operatorname{Fact}^+ x$ for the set of non-empty factors of $x$. Say that $x$ belongs to $\mathcal P$ if there is…