Sharp Bounds for Toral Eigenfunctions and Threefold Additive Energy on Circles

For the lattice circle Pₙ=²:|z|²=n\, we prove that the threefold additive energy J₃(Pₙ)≤ C|Pₙ|³, with an absolute constant C. This establishes the optimal bound suggested by Krishnapur--Kurlberg--Wigman and Bombieri--Bourgain. We also prove a uniform weak L⁶ bound for L²-normalized eigenfunctions on the square torus, and hence uniform Lᵖ bounds for every 1≤ p<6. This extends the classical L⁴ bounds of Cooke and Zygmund and establishes Bourgain's conjecture in this range. Applications include mean-square convergence for L6/5 observables and a logarithm-free remainder in the nodal length variance of arithmetic random waves.