5 problems

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 $Ck$…

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…

Let $N_\alpha(k,k)$ denote the quantity defined in the paper as the relevant threshold for words over an alphabet of size $\alpha$, avoiding $k$-powers and $k$-anti-powers. Alphabe…

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…