Erdős Problem 66

Reference: erdosproblems.com/66

theorem Erdos66.erdos_66 : (∃ (A : Set ℕ) (c : ℝ), c ≠ 0 ∧ Filter.Tendsto (fun (n : ℕ) => ↑(AdditiveCombinatorics.sumRep A n) / Real.log ↑n) Filter.atTop (nhds c)) ↔ sorry

Is there and $A \subset \mathbb{N}$ is such that $$\lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n}$$ exists and is $\ne 0$?