Additive energies of subsets of discrete cubes

Abstract: For a positive integer $n \geq 2$, 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 $|A|^{t_n}$. Trivially we have $t_n \leq 3$ and $$ t_n \geq 3 - \log_n\frac{3n^3}{2n^3+n} $$ 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: $$ 3 - (1+o_{n\rightarrow\infty}(1)) \log_n \frac{3\sqrt{3}}{4} \leq t_n \leq 3 - \log_n(1+c), $$ where $c>0$ is an absolute constant.