On an asymmetric additive energy inequality

Abstract

Let $d \geq 1$ be an integer, $G$ be an abelian group and $ν, w_1, \dots, w_{2d}: G \to [0, \infty)$ be functions with finite, non-empty supports. Define the generalised additive energy \[ E_{2d, ν}(w_1, \dots, w_{2d}) = \sum_{y,y' \in G}\sum_{a_1, \dots, a_{2d} \in G } w_1(a_1) \dots w_{2d}(a_{2d}) ν(y) ν(y') 1_{\sum_{i=1}^d (a_i - a_{i+d}) = y-y'} .\] Moreover, for every $1 \leq i \leq 2d$, let $E_{2d, ν}(w_i) = E_{2d, ν}(w_i, \dots, w_i)$. A standard Fourier analytic argument delivers the estimate \[ E_{2d,ν}(w_1, \dots, w_{2d}) \leq \prod_{1 \leq i \leq 2d} E_{2d, ν}(w_i)^{1/2d}.\] 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.

Disclosure

“boldface to denote vectors z = (z1 , z2 , . . . , zk ) ∈ Z k . Acknowledgements. The author would like to thank Anurag Sahay and Ben Green for helpful comments. The author is supported by a Leverhulme Early Career Fellowship ECF-2025-148. Copilot was used in the process of proving Proposition 1.3; all other ideas were generated by the author. This article is written entirely by the author. 2. A Fourier analytic proof of Theorem 1.1 For every f : G → C, let s”

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2026-08-23T19:10:14.826820Z

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