15 problems

For each integer $k\geq 1$, let ${\bf x}_k$ be the infinite word defined as the limit of the locally catenative sequence described above, and let $\rho_{{\bf x}_k}(n)$ denote its s…

Let ${\mathbf G}=\gamma^\omega(0)$, where $\gamma=[01,2,02]$, let $h$ be a morphism in Table, and let $h^R$ denote the reversed morphism used in that table. Let $\mu$ be the critic…

Let ${\mathbf G}=\gamma^\omega(0)$, where $\gamma=[01,2,02]$, and let $\mu$ be its critical exponent. For each morphism $h$ in Table, consider the infinite word $h({\mathbf G})$. M…

For each integer $k\geq 1$, let ${\bf x}_k$ be the infinite word defined as the limit of the locally catenative sequence described above. The critical exponent conjecture for…

A game is morphic if its $P$-positions can be retrieved from a recoding of a fixed point of a morphism. The $3$- and $4$-Queen Dees are the corresponding higher-dimensional Queen D…

Let $(a_n,b_n)$ denote the $P$-positions of the $2$-Queen Dee, and let $\mathbf t$ be the associated infinite word. Fokkink and Rigo's conjecture. The $P$-positions of the $2$-Quee…

Let $x^{(00)}$ be the Thue–Morse two-block fixed point, and let the frequency of $1$ mean the limiting proportion of positions occupied by $1$ in its prefixes, if that limit exists…

Let $x^{(00)}$ be the fixed point of the Thue–Morse two-block substitution. A word is uniformly recurrent when every finite factor occurring in $x^{(00)}$ occurs there with bounded…

Let $x^{(00)}$ be the fixed point of the Thue–Morse two-block substitution, and let $L_{\rm\sc TM}$ be the language of all finite words occurring in $x^{(00)}$. A set of binary wor…

Let $A(n)=\lfloor n\sqrt{2}\rfloor$ and let $AA$ denote the iterated Beatty sequence obtained by applying $A$ to the terms of $A$. A morphic representation of $AA$ is given on an a…

A morphic word $w$ is an infinite word generated by iterating a morphism; an antipower is a word formed by concatenating pairwise distinct blocks of equal length. Berger–Defant's c…

Linear-length anti-power conjecture. There is a constant $C=C(W)$ such that, for all positive integers $i$ and $k$, $W$ contains a $k$-anti-power with blocks of length at most $Ck$…

Classification conjecture. Let $\mathbf u$ be an aperiodic binary generalized pseudostandard word, not standard Sturmian, being a fixed point of a primitive morphism. Then

A fixed point of a primitive morphism is an infinite word ${\bf u}$ fixed by a morphism whose incidence matrix is primitive. Its palindromic defect $D({\bf u})$ measures the failur…

Let ${\bf u}$ be a fixed point of a primitive morphism $\varphi$. The defect of ${\bf u}$ is the supremum, over all finite factors of ${\bf u}$, of the difference between the facto…