2026/07/28 by Akshat Mudgal
#math.NT #math.CO
Let d ≥ 1 be an integer, G be an abelian group and ν, w1, …, w2d: G → [0, ∞) be functions with finite, non-empty supports. Define the generalised additive energy E2d, ν(w1, …, w2d) = ∑y,y' ∈ G∑_a1, …, a2d ∈ G w1(a1) … w2d(a2d) ν(y) ν(y') 1_∑i=1d (ai - ai+d) = y-y' . Moreover, for every 1 ≤ i ≤ 2d, let E2d, ν(wi) = E2d, ν(wi, …, wi). A standard Fourier analytic argument delivers the estimate E2d,ν(w1, …, w2d) ≤ ∏1 ≤ i ≤ 2d E2d, ν(wi)1/2d. In this note, we present a purely combinatorial proof of the above inequality. In particular, our proof does not use any Fourier or spectral analysis and relies on repeated applications of Cauchy--Schwarz inequality combined with a discrete convexity extension type argument. We also record a variation of this upper bound in the non-abelian setting via spectral inequalities following work of Hatami on graph norms, as well as a relevant sumset analogue obtained via iterative applications of the Plünnecke--Ruzsa inequality.