Is any subset of the naturals a semi-linear set?

Question

A subset $X$ of $\mathbb{N}^n$ is linear if it is in the form:

$u_0 + \langle u_1,...,u_m \rangle = \{ u_0 + t_1 u_1 + ... + t_m u_m \mid t_1,...,t_n \in \mathbb{N}\}$ for some $u_0,...,u_m \in \mathbb{N}^n$

$X$ is semilinear if it is the union of finitely many linear subsets. So my question is: Is any subset of the natural numbers a semi-linear set, i.e., can we express any subset of the natural numbers as the finite union of linear sets?

Answer 1

Consider the set $\{2^n:n\in\mathbb{N}\}$, i.e., the powers of $2$. Note that the inter-element differences are all different. Therefore, the only linear subsets are singleton elements. Hence, this is not a finite union of linear sets.

Answer 2

There are only countably many semilinear sets, but uncountably many subsets of $\mathbb{N}^n$.