Additive energies on spheres

Abstract

In this paper, we study additive properties of finite sets of 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 of lattice points ( x 1 , ⋯ , x d ) ∈ Z d $(x_1, \dots , x_d) \in \mathbb {Z}^d$ satisfying x 1 2 + ⋯ + x d 2 = m $x_1^2 + \dots + x_{d}^2 = m$ . When d = 4 $d=4$ , we prove threshold breaking bounds for the additive energy of A $A$ , that is, we show that there are at most O ε ( m ε | A | 2 + 1 / 3 − 1 / 2766 ) $O_{\epsilon }(m^{\epsilon }|A|^{2 + 1/3 - 1/2766})$ solutions to the equation a 1 + a 2 = a 3 + a 4 $a_1 + a_2 = a_3 + a_4$ , with a 1 , ⋯ , a 4 ∈ A $a_1, \dots , a_4 \in A$ . 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 Z 4 $\mathbb {Z}^4$ , since the threshold bound is sharp in the latter case. We also obtain variants of this estimate when d = 3 $d=3$ , 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.