Let $C>0$. Is it true that the set of integers of the form$$n=b_1+\cdots +b_t\text{ with }{b}_1<\cdots <b_t$$where $b_i=2^{k_i}{3}^{l_i}$ for $1\le i\le t$ and $b_t\le C{b}_1$ has density $0$?

AMS 11 · solved (literature)

theorem erdos_845 :
    answer(False) ↔
      ∀ᵉ (C : ℝ) (hC : 0 < C),
        let f : ℕ × ℕ → ℕ := fun (k, l) ↦ 2 ^ k * 3 ^ l
        { ∑ x ∈ B, f x | (B : Finset (ℕ × ℕ)) (h : B.Nonempty)
          (hB : B.sup f ≤ C * B.inf' h f) }.HasDensity 0