2 problems

Let $k$ and $\ell$ be multiplicatively independent positive integers. A subset of $\mathbb{N}$ is sparse if its counting function is bounded by $O((\log x)^d)$ for some $d$. Let…

Let $k\geq 2$ and let $L$ be a regular language of base-$k$ representations of pairs whose quotient set is … A representation in $L$ is a word encoding a pair of nonnegative intege…