Bourgain Λ(r)-set predictions for arithmetic paraboloids and spheres

Establish that the integer paraboloid Γ = {(n₁,n₂,n₁²+n₂²) : n₁,n₂ ∈ ℤ} is a Λ(r)-set for every r < 4 and that each nonzero integer sphere S_R = {n ∈ ℤ³ : n₁²+n₂²+n₃² = R} is a Λ(r)-set for every r < 6.

Background

The paper explains that the finite-field restriction conjecture would have consequences for additive harmonic-analysis properties of two integer sets: the discrete paraboloid Γ and the integer sphere S_R for R ≠ 0. Specifically, the cited prediction assigns the ranges r < 4 and r < 6, respectively.

These statements are presented as predictions by Bourgain rather than established results. The wider range for the sphere is attributed to its greater arithmetic sparsity.