Right, so the Erdős–Turán conjecture for additive bases (of order 2) says, with the usual notations, that $\sup r_B (n) = \infty$. Let’s look instead at the average number of representations, i.e.: the function

$$F(N) = \frac1N \sum_{n=1}^{N} r_B (n)$$

This seems entirely natural to me, indeed more natural than looking at $(\lim)\sup r_B (n)$, if the idea is to demonstrate that a basis must have some “thickness”. There are examples known where $\sup F(N) < \infty$, indeed any so-called “thin” basis has this property. Recall that a basis is “thin” if there exists $c > 0$ such that the $k$th element of the basis is at least $c \cdot k^2$. Now an obvious question to ask is whether $\limsup F(N)$ can equal one? (It must be at least one if $B$ is a basis.) I have searched the literature in vain for an answer. In particular, I have not found any result on thin bases which directly translates into an answer to this question, though that may because I am too stupid to see some connection. Any comments most welcome.