A generalization of the Hilbert–Waring theorem

Let
$$\mathbf Z(m,r)=\biggl\{n\mid n\neq\sum^s(n_i^r-m^r) \quad\text{for all}\quad s\geq1,n_i\geq m\biggr\}$$
and
$$\mathbf N(m,r,s)=\biggl\{n\mid n\neq\sum^s n_i^r,n_i\geq m,n> s\cdot m^r\biggr\}.$$
Then, for any $m\geq0$, $r\geq2$ there exist the number, $g(m,r)$, and the finite invariante set, $\mathbf Z(m,r)$, such that for any $s\geq g(m,r)$
$$\mathbf N(m,r,s)=\{s\cdot m^r+z\mid z\in\mathbf Z(m,r)\},$$
If $m=0$ then we obtain the classical Hilbert–Waring theorem.