Sparse automatic set and zero-density automatic set intersection conjecture

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 $X\subseteq\mathbb{N}$ be sparse and $k$-automatic, and let $Y\subseteq\mathbb{N}$ be zero-density and $\ell$-automatic. Sparse–zero-density intersection conjecture. The intersection $X\cap Y$ is finite. The heuristic in the source uses the polynomial decay of the density of $Y$ together with the rapidly growing enumeration of the sparse set $X$; the conjecture is presented as beyond current methods in number theory.