The Averaged Representation Count of an Additive Basis

For $A \\subseteq \\mathbb{N}$ let $r_A(n)$ count the ordered pairs from $A$ summing to $n$. The partial sum $\\sum_{n<N} r_A(n)$ is not an approximation to anything: it counts exactly the pairs from $A$ whose sum lies below $N$, and both entries of such a pair are themselves below $N$. Hence the partial sum is at most $|A \\cap [0,N)|^2$. If $A$ is a basis of order two then the same partial sum is at least $N - N_0$, because every large $n$ has a representation. The two bounds together give the square-root density estimate $N - N_0 \\le |A \\cap [0,N)|^2$, and they squeeze the average $N^{-1}\\sum_{n