Erdős Problem #337

Let \(A\subseteq \mathbb{N}\) be an additive basis (of any finite order) such that \(\lvert A\cap \{1,\ldots,N\}\rvert=o(N)\). Is it true that \[ \lim_{N\to \infty}\frac{\lvert (A+A)\cap \{1,\ldots,N\}\rvert} {\lvert A\cap \{1,\ldots,N\}\rvert}=\infty? \] The answer is no, and a counterexample was provided by Turjányi [Tu84]. This was generalised (to the replacement of \(A+A\) by the \(h\)-fold sumset \(hA\) for any \(h\geq 2\)) by Ruzsa and Turjányi [RT85]. "Additive basis" is `Set.IsAsymptoticAddBasis`: some finite \(h\) has \(hA\) containing every sufficiently large integer. The exact notion `Set.IsAddBasis`, which asks that \(hA\) be all of \(\mathbb{N}\), would force \(0, 1 \in A\) and is not the class these results are about. The linked file states the basis hypothesis as `∃ N₀, Set.Ici N₀ ⊆ iterated_sumset A k` and indexes both counting functions by a real \(x\) through \(\lfloor x\rfloor\), where the counting functions here are indexed by \(N : \mathbb{N}\).