Complexity-sensitive additive energy and off-diagonal Young inequalities on bounded-degree algebraic varieties

Abstract: 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 $σ(V)=2m-\dim\overline{V-V}<sup>{\mathrm{Zar}}$ 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 $Λ<em>{a,R}(X;V)$ and prove $$E(X)\ll Λ</em>{a,R}(X;V)<sup>{3-a}|X|<sup>{a+\varepsilon}$$. This recovers the line-concentration theorem of Jing and Wu for algebraic surfaces in $\mathbb{R}<sup>3$. For codimension-two quadratic threefolds $${(u,Q_1(u),Q_2(u))\in\mathbb{R}<sup>3}\subset\mathbb{R}<sup>5$$ with positive-definite $Q_1$ and simple generalized spectrum, we prove the sharp estimate $E(X)\ll_{\varepsilon}|X|<sup>{2+\varepsilon}$ without a flag loss. Hereditary versions of these estimates imply weighted $L<sup>4$ 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: $$J_3(P)\ll_{\varepsilon}κ(P)<sup>2|P|<sup>{3+\varepsilon}$$, with matching examples at every power scale.