If $g_3(n)$ is the largest size of $A \subseteq [1,n]$ with fewer than three representations of every product $a_1 a_2$, does its conjectured second-order normalized term converge? The candidate proof gives an explicit limit constant.

Let $k\geq 2$ and let $g_k(n)$ be the largest possible size of $A\subseteq \{1,\ldots,n\}$ such that every $m$ has $<k$ solutions to $m=a_1a_2$ with $a_1<a_2\in A$. Is it true that $$g_3(n)=\frac{\log\log n}{\log n}n+(c+o(1))\frac{n}{\log n}$$ for some constant $c$?

The answer is yes: the rescaled error normalizedError converges.