Additive energies on spheres

In this paper, we study additive properties of finite sets of lattice points on spheres in $3$ and $4$ dimensions. Thus, given $d,m \in \mathbb{N}$, let $A$ be a set of lattice points $(x_1, \dots, x_d) \in \mathbb{Z}^d$ satisfying $x_1^2 + \dots + x_{d}^2 = m$. When $d=4$, we prove threshold breaking bounds for the additive energy of $A$, that is, we show that there are at most $O_{\epsilon}(m^{\epsilon}|A|^{2 + 1/3 - 1/1392})$ solutions to the equation $a_1 + a_2 = a_3 + a_4,$ with $a_1, \dots, a_4 \in A$. This improves upon a result of Bourgain and Demeter, and makes progress towards one of their conjectures. A further novelty of our method is that we are able to distinguish between the case of the sphere and the paraboloid in $\mathbb{Z}^4$, since the threshold bound is sharp in the latter case. We also obtain variants of this estimate when $d=3$, where we improve upon previous results of Benatar and Maffucci concerning lattice point correlations. Finally, we use our bounds on additive energies to deliver discrete restriction type estimates for the sphere.