Congruence classes and maximal nonbases

The set $A$ is an asymptotic nonbasis of order $h$ for an additive abelian group $X$ if there are infinitely many elements of $X$ not in the $h$-fold sumset $hA$. For all $h\ge 2$, this paper constructs new classes of asymptotic nonbases of order $h$ for $\mathbf{Z}$ and for ${\mathbf{N}}_0$ that are not subsets of maximal asymptotic nonbases.