8 problems

Let $\mathbf{t}=01101001\cdots$ be the Thue–Morse word, and let $c_{\mathbf{t}}(n)$ denote the number of length-$n$ factors of $\mathbf{t}$, counting factors that differ only by a…

Let $\gamma_j(k)$ be the function measuring the least relevant anti-power length of order $k$ for the shift parameter $j$ in the Thue–Morse word. For each nonnegative integer $j$,…

Let $b$ be the Thue–Morse word, and let $AP_j(\mathbf{t},k)$ denote the set of lengths of anti-power $j$-fixes of order $k$ in $b$. For fixed $j,k\in\mathbb{Z}^{\geq 0}$ with…

Growth conjecture. There exists a sequence $(a_i)$ converging to $0$ such that, whenever $n$ lies between $3\cdot 2^i$ and $2^{i+1}(1-a_i)$,

Defant's conjecture.

Let ${\bf t}$ be the Thue–Morse word, and let $\gamma(k)$ denote the least positive integer $n$ such that the prefix of ${\bf t}$ of length $nk$ is a $k$-anti-power. Then the asymp…

Let $t$ be the Thue–Morse word, and let $\mathpzc{A}_t$ denote its privileged complexity function, whose value at each length counts the privileged factors of $t$ of that length. P…

Let $t$ be the Thue–Morse word, and let $\mathpzc{A}_n(t)$ denote the number of privileged factors of $t$ of length $n$. A zero gap is a consecutive interval of lengths on which…