AI 中文总结
该研究针对有界度实代数簇上的有限集合推导加性能量估计与加权杨不等式，还原Jing和Wu的直线集中定理，证明余维2二次三维簇的精确能量估计及Cushman-Demeter-Wu定理的转向复杂度扩展。
AI 中文摘要
我们针对有界度实代数簇上的有限集合，推导加性能量估计与加权杨不等式。对于不可约m维簇V，定义σ(V)=2m−dim\bar{V−V}^{Zar}，α(V)=max{2,1+2σ(V)/m}。对每个a∈[α(V),3)，定义有限度平移分块标志参数Λ_{a,R}(X;V)，并证明E(X)≪Λ_{a,R}(X;V)^{3−a}|X|^{a+ε}，该结果可还原Jing与Wu关于R^3中代数曲面的直线集中定理。对于具有正定Q₁与简单广义谱的余维2二次三维簇{(u,Q₁(u),Q₂(u))∈R³}⊂R⁵，我们证明无标志损失的精确估计E(X)≪_ε|X|^{2+ε}。这些估计的遗传版本蕴含加权L⁴限制界与非对角杨不等式，近对角阈值处的精确区域为1≤p,q≤2且p⁻¹+q⁻¹≥1。我们还证明了Cushman-Demeter-Wu定理的精确转向复杂度扩展：J₃(P)≪_εκ(P)²|P|^{3+ε}，且在每个幂尺度上均有匹配实例。
英文摘要
We develop additive-energy estimates and weighted Young inequalities for finite sets on bounded-degree real algebraic varieties. For an irreducible $m$-dimensional variety $V$, let $σ(V)=2m-\dim\overline{V-V}^{\mathrm{Zar}}$ and $α(V)=\max{2,1+\frac{2σ(V)}{m}}$. For every $a\in[α(V),3)$ we define a finite-degree translation-partition flag parameter $Λ_{a,R}(X;V)$ and prove $E(X)\ll Λ_{a,R}(X;V)^{3-a}|X|^{a+\varepsilon}$. This recovers the line-concentration theorem of Jing and Wu for algebraic surfaces in $\mathbb{R}^3$. For codimension-two quadratic threefolds ${(u,Q_1(u),Q_2(u))\in\mathbb{R}^3}\subset\mathbb{R}^5$ with positive-definite $Q_1$ and simple generalized spectrum, we prove the sharp estimate $E(X)\ll_{\varepsilon}|X|^{2+\varepsilon}$ without a flag loss. Hereditary versions of these estimates imply weighted $L^4$ restriction bounds and off-diagonal Young inequalities; at the near-diagonal threshold the sharp region is $1\le p,q\le 2$ and $p^{-1}+q^{-1}\ge 1$. We also prove a sharp turning-complexity extension of the Cushman-Demeter-Wu theorem: $J_3(P)\ll_{\varepsilon}κ(P)^2|P|^{3+\varepsilon}$, with matching examples at every power scale.
Comments26 pages, no figures. An accompanying formalization project is in progress at https://github.com/hxypqr/complexity-sensitive-additive-energy
