Additive energies of subsets of discrete cubes

For a positive integer $n\ge 2$, define $t_n$ to be the smallest number such that the additive energy $E(A)$ of any subset $A\subset \{0,1,\cdots ,n-1{\}}^d$ and any $d$ is at most $\mathrm{\mid }A{\mathrm{\mid }}^{t_n}$. Trivially we have $t_n\le 3$ and $$t_n\ge 3-{\log }_n\frac{3n^3}{2n^3+n}$$ by considering $A=\{0,1,\cdots ,n-1{\}}^d$. In this note, we investigate the behavior of $t_n$ for large $n$ and obtain the following non-trivial bounds: $$3-(1+o_{n\to \mathrm{\infty }}(1)){\log }_n\frac{3\sqrt{3}}{4}\le t_n\le 3-{\log }_n(1+c),$$ where $c>0$ is an absolute constant.