Additive energies on spheres

• Published in: Journal of the London Mathematical Society 2022-12, Vol.106 (4), p.2927-2958
• Main Author: Mudgal, Akshat
• Format: Article
• Language: English
• ISSN: 0024-6107; 1469-7750
• DOI: 10.1112/jlms.12652

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 (x1,⋯,xd)∈Zd$(x_1, \dots , x_d) \in \mathbb {Z}^d$ satisfying x12+⋯+xd2=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 Oε(mε|A|2+1/3−1/2766)$O_{\epsilon }(m^{\epsilon }|A|^{2 + 1/3 - 1/2766})$ solutions to the equation a1+a2=a3+a4$a_1 + a_2 = a_3 + a_4$, with a1,⋯,a4∈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 Z4$\mathbb {Z}^4$, 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.