Let $A=\{a_1<\cdots <a_k\}$ be a finite set of positive integers and extend it to an infinite sequence $\overline{A}=\{a_1<a_2<\cdots \}$ by defining $a_{n+1}$ for $n\ge k$ to be the least integer exceeding $a_n$ which is not of the form $a_i+a_j$ with $i,j\le n$. Is it true that the sequence of differences $a_{m+1}-a_m$ is eventually periodic?

AMS 11 · open (literature)

theorem erdos_341 :
    answer(sorry) ↔
      ∀ (a : ℕ → ℤ),
        (∀ᶠ n in atTop,
          IsLeast { x | a n < x ∧ x ∉ { a i + a j | (i ≤ n) (j ≤ n) } } (a (n + 1))) →
        let d := fun i ↦ a (i + 1) - a i
        ∃ p > 0, ∀ᶠ m in atTop, d (m + p) = d m