Computational Evidence for Erd˝os Problem #158 via the Greedy B2[2] Construction

Erd\H{o}s Problem \#158 \cite{ErdosProblems158} asks whether every infinite set $A\subset\mathbb N$ satisfying $r_A(n)\le 2$ for all $n$ (i.e. a $B_2[2]$ set) must satisfy
\[
\liminf_{N\to\infty}\frac{|A\cap\{1,\dots,N\}|}{\sqrt N}=0.
\]
The analogous statement for Sidon sets ($B_2[1]$) is known to hold (see, for example, \cite{Ruzsa04}).
A natural counterexample candidate for \#158 is the classic greedy $B_2[2]$ construction.
This note records verified computations for the first $2000$ greedy elements, reaching
$a_{2000}=7{,}445{,}662$, and reports checkpoint values of the normalized counting function.
In the computed range the ratio $|A\cap[1,N]|/\sqrt N$ decreases from $1.8974$ at $N=10$ to $0.7330$ at $N=7{,}445{,}662$.
A reference implementation and a performance-oriented skeleton (with checkpointing recommendations) are included to facilitate larger runs up to $N=10^{10}$.