Power-Saving Lower Bounds for Additive Bases of Polynomial Sequences

Let h ≥ 2 be fixed, let f be a natural-valued polynomial of degree d ≥ 2, and suppose that A ⊆ ℕ satisfies f(ℕ) ⊆ hA. We prove |A ∩ [0,X]| ≥ X4/(3hd)−o(1) when h is even, and |A ∩ [0,X]| ≥ X4/((3h+1)d)−o(1) when h is odd. Both exponents are strictly larger than the elementary exponent 1/(hd). This gives an affirmative qualitative answer, for every fixed h ≥ 2, to Problem 2.8 from the 2004 AIM problem list on recent trends in additive combinatorics. The argument combines a self-contained graph-path form of an Erdős–Newman two-basis estimate with a divisor bound for repeated differences of polynomial values. For odd h, one summand is first frozen on a large fiber and the remaining summands are then split into two equal blocks. No optimality of the displayed exponents is asserted. Research status: Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-use disclosure: AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text.