A subset $B$ is called an (asymptotic) additive basis of order $2$ if every sufficiently large natural number $n$ can be written as the sum of at most $2$ elements of $B.$
How small/sparse can such sets be asymptotically?
So for example, let $b_n$ be the $n-th$ element of $B$, when $B$ is written as an increasing sequence of integers, rather than a set. Do we have, for example:
$$\ \lim_{n\to\infty} \frac{b_n}{n^2} = 0\ ? $$
Are there any known good upper bounds of $O(b_n)?\ $
I came up with this question while thinking about the Goldbach conjecture, and this led me to read wikipedia's articles on Erdős–Turán conjecture on additive bases, although I believe my question is tangential. Sidon sequences may also be relevant.
Edit: relevant question $1$ $\qquad$ Relevant question 2
"Thin basis" definition found here:
The set $A$ of nonnegative integers is called a basis of order $h$ if every nonnegative integer can be represented as the sum of exactly $h$ not necessarily distinct elements of $A$. An additive basis $A$ of order $h$ is called thin if there exists $c>0$ such that the number of elements of $A$ not exceeding $x$ is less than $xc^{1/h}$ for all $x\geq 1.$
This paper shows how to construct sets such that $O(b_n) = n^2.$
Am I right then in thinking this is the best upper bound for $O(b_n)$ asymptotically? For example, we can make a similar argument to this simple counting/combinatorial one to conclude that $O(b_n) = n^3$ is impossible?
