Sublinear Michaux-Villemaire over $(\mathbb{N}, +, k^\mathbb{N})$

In 1996, Michaux and Villemaire showed that any nontrivial expansion of $(\mathbb{N}, +)$ defines new unary sets. We conjecture that the same holds over the structure $(\mathbb{N}, +, k^\mathbb{N})$, and we prove a partial result in this direction. To do so, we first show that every nonempty sparse