Near diagonal additive energy bound for points on algebraic surfaces

Abstract

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.