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

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 $\sigma (V)=2m-\dim {\overline{V-V}}^{\mathrm{Z}\mathrm{a}\mathrm{r}}$ and $\alpha (V)=\max 2,1+\frac{2\sigma (V)}{m}$. For every $a\in [\alpha (V),3)$ we define a finite-degree translation-partition flag parameter ${\mathrm{\Lambda }}_{a,R}(X;V)$ and prove $E(X)\ll {\mathrm{\Lambda }}_{a,R}(X;V{)}^{3-a}\mathrm{\mid }X{\mathrm{\mid }}^{a+\epsilon }$. 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 $E(X){\ll }_{\epsilon }\mathrm{\mid }X{\mathrm{\mid }}^{2+\epsilon }$ 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: $J_3(P){\ll }_{\epsilon }\kappa (P{)}^2\mathrm{\mid }P{\mathrm{\mid }}^{3+\epsilon }$, with matching examples at every power scale.