An Extension of the Guth–Maynard Additive‑Energy Estimate to 𝑁 > 𝑇 2 / 3

Abstract This paper extends the Guth–Maynard additive‑energy estimate for large values of Dirichlet polynomials. Guth and Maynard proved that for T3/4≤N≤T, the additive energy of a large‑value set satisfies a three‑term bound. We show that the identical bound remains valid in the larger range T2/3<N≤T.

The proof retains the unreduced large‑gcd Cauchy product from Guth–Maynard’s Section 11 rather than collapsing its mixed term into a single monomial. In the small‑cardinality branch, the mixed term is interpolated between two existing target monomials using the exact inequality

As a consequence, the Third Guth–Maynard relation in the ANTEDB large‑value‑energy region extends from 1<τ<4/3 to 1<τ<3/2, including the endpoint τ=3/2 by a subdivision argument. We also identify 2/3 as the precise barrier for the coarse monomial‑domination mechanism used here; this is not a sharpness statement for the theorem itself. At one rational point in the new range we obtain a direct energy‑envelope gain of 23/400.

The scope of the paper is deliberately narrow. We do not claim a new global zero‑density estimate, an improvement of the Guth–Maynard cardinality theorem, or any statement concerning the Riemann Hypothesis. The contribution is a clean range extension of a published energy estimate, together with its translation into the Analytic Number Theory Exponent Database (ANTEDB).

Keywords Dirichlet polynomials; large values; additive energy; Guth–Maynard method; ANTEDB; large‑value‑energy region

Mathematical Status Main theorem: Guth–Maynard additive‑energy estimate extended to N>T2/3. Global zero‑density bounds: not claimed. Riemann Hypothesis: not obtained.