4 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 $P=\{2^n:n\ge 0\}$ be the set of powers of two, and let $T$ be the set of natural numbers whose ternary expansions omit the digit $2$. The set $P$ is $2$-automatic and $T$ is…

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…

Let $k,l\geq 2$ be multiplicatively independent integers. A set $S$ of rational numbers is simultaneously $k$- and $l$-automatic if it is accepted by finite automata in both bases.…