关于一个非对称加法能量不等式
On an asymmetric additive energy inequality
浏览论文内容
中文总结 AI 辅助
研究关于广义加法能量不等式，给出其纯组合证明，不依赖傅里叶或谱分析，通过柯西 - 施瓦茨不等式等论证，还记录了非阿贝尔情形变体及和集类似结果。
中文摘要 AI 辅助
设\(d \geq 1\)为整数，\(G\)为阿贝尔群，\(\nu, w_1, \dots, w_{2d}: G \to [0, \infty)\)为具有有限非空支撑集的函数。定义广义加法能量\(E_{2d, \nu}(w_1, \dots, w_{2d})\)。标准傅里叶分析论证给出一个估计。本文给出上述不等式的纯组合证明，不使用傅里叶或谱分析，依赖柯西 - 施瓦茨不等式的重复应用及离散凸性扩展型论证。还记录了非阿贝尔情形下的一个变体及相关和集类似结果。
英文摘要
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.
