New Bounds for the Discrete Fourier Restriction to the Sphere in 4D and 5D

We prove explicit upper bounds for local densities in the Siegel mass formula for systems of quadratic equations. Using this we derive sharp estimates for the additive energy of the lattice points on the 4D and 5D sphere. When combined with incidence geometry this leads to an improvement in the range for the discrete Fourier restriction for the sphere in 4D and 5D.