AI 中文总结
该论文针对$\boldsymbol{R}^3$上不可约、次数不低于2的多项式定义的代数曲面，证明了不集中于仿射直线的点集的加性能量上界
AI 中文摘要
设$F:\boldsymbol{R}^3\to\boldsymbol{R}$为$\boldsymbol{R}$上不可约且次数$\text{deg}(F)\boldsymbol{\times}2$的多项式，证明对不集中于仿射直线的任意有限子集$X\boldsymbol{\times}Z(F)$，加性能量$E(X)=\boldsymbol{\times}\text{满足}\boldsymbol{\times}X^4:a+b=c+d\boldsymbol{\times}\boldsymbol{\times}_{\text{deg}(F),\boldsymbol{\times}}(\boldsymbol{\times}X)^{2+\boldsymbol{\times}}$
英文摘要
Let $F:\mathbb{R}^3\to\mathbb{R}$ be a polynomial that is irreducible over $\mathbb{R}$ with $\text{deg} F\geq2$. We prove that, for any finite $X\subset Z(F)$ that does not concentrate on affine lines, \[ E(X)=\#\{(a,b,c,d)\in X^4: a+b=c+d\}\ll_{\text{deg} F,\,ε}(\# X)^{2+ε}. \] In particular, this answers a question of Bourgain and Demeter concerning finite subsets of the unit sphere.
CommentsMinor clarifications to the introduction
