Erdős Problem #477

Is there a polynomial \(f:\mathbb{Z}\to \mathbb{Z}\) of degree at least \(2\) and a set \(A\subset \mathbb{Z}\) such that for any \(n\in \mathbb{Z}\) there is exactly one \(a\in A\) and \(b\in \{ f(k) : k\in\mathbb{Z}\}\) such that \(n=a+b\)? The answer is yes, contrary to the expectation of Erdős and Graham: such an `A` exists whenever \(f(n) = n^d\) for even \(d \ge 6\). The linked formal proof (Codex, following Price's exposition) exhibits a complement of \(\{k^6 : k \in \mathbb{Z}\}\); it states uniqueness as `∃! p : ℤ × ℤ, p.1 ∈ A ∧ p.2 ∈ B ∧ p.1 + p.2 = n` and the degree condition as `2 ≤ f.natDegree`.