Additive energies of subsets of discrete cubes

For a positive integer n ≥ 2, define tₙ to be the smallest number such that the additive energy $E(A)$ of any subset A ⊂ \0,1,⋯,n-1\ᵈ and any d is at most |A|tₙ. Trivially we have tₙ ≤ 3 and tₙ ≥ 3 - logₙ3n³/2n³+n by considering A = \0,1,⋯,n-1\ᵈ. In this note, we investigate the behavior of tₙ for large n and obtain the following non-trivial bounds: 3 - (1+on→∞(1)) logₙ 3√34 ≤ tₙ ≤ 3 - logₙ(1+c), where $c>0$ is an absolute constant.