2 problems

Hansel–Safer conjecture. Assuming the four exponentials conjecture, if $S$ is both $\alpha$- and $\beta$-recognizable, then $S$ is eventually periodic.

Let $k,\ell\in\mathbb{N}$ be multiplicatively independent, meaning that no positive powers of $k$ and $\ell$ are equal. Let $\mathbf a=(a_n)_{n\in\mathbb{N}_0}$ be an integer seque…