Additive bases of positive integers and related problems

Let $A$ be an infinite set of nonnegative integers and let $k \geq 2$ be an integer. We investigate the relation between the number of representations of an integer $n$ by sums of the form $a_1+ \dots+a_k$, where $a_1, \dots, a_k \in A$,and the size of $A$. Some related problems are also considered.