Additive energies on spheres

In this paper, we study additive properties of finite sets lattice points on spheres in three and four dimensions. Thus, given d , m ? N $d,m \in \mathbb {N}$ let A $A$ be a set ( x 1 ? ) Z $(x_1, \dots x_d) {Z}^d$ satisfying 2 + = $x_1^2 x_{d}^2 m$ . When 4 $d=4$ prove threshold breaking bounds for the energy that is, show there are at most O ? | / 3 ? 2766 $O_{\epsilon }(m^{\epsilon }|A|^{2 1/3 - 1/2766})$ solutions to equation $a_1 a_2 a_3 a_4$ with $a_1, a_4 A$ This improves upon result Bourgain Demeter, makes progress towards one their conjectures. further novelty our method is able distinguish between case sphere paraboloid $\mathbb {Z}^4$ since bound sharp latter case. We also obtain variants estimate when $d=3$ where improve previous results Benatar Maffucci concerning point correlations. Finally, use energies deliver discrete restriction-type estimates sphere.