Additive energies of subsets of discrete cubes

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 \begin{equation*} t_n \geq 3 - \log_n\frac{3n^3}{2n^3+n} \end{equation*} by considering $A = \{0,1,\cdots,n-1\}^d$ . In this note, we investigate the behaviour of t n for large n and obtain the following non-trivial bounds: \begin{equation*} 3 - (1+o_{n\rightarrow\infty}(1)) \log_n \frac{3\sqrt{3}}{4} \leq t_n \leq 3 - \log_n(1+c), \end{equation*} where c > 0 is an absolute constant.