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

This problem is discussed under Problem 7 on Green's open problems list.

The answer is no: Li [Li26] (with GPT-5.6 Sol) showed that the greedy extension of the seed set $A = {1, 2, 3, 5, 7, 13, 22, 27, 28, 32, 36, 40, 47, 48, 52, 63, 71, 77, 81, 89, 97}$ has a sequence of differences that is not eventually periodic. The linked formal proof exhibits such a sequence a : ℕ → ℕ (strictly increasing, with the greedy rule holding from some index on, and with n ↦ a (n + 1) - a n not eventually periodic); casting it to ℤ gives a counterexample to the statement below.