Additive energies on discrete cubes

We prove that for $d\ge 0$ and $k\ge 2$, for any subset $A$ of a discrete cube $\{0,1{\}}^d$, the $k-$higher energy of $A$ (the number of $2k-$tuples $(a_1,a_2,\dots ,a_{2k})$ in $A^{2k}$ with $a_1-a_2=a_3-a_4=\cdots =a_{2k-1}-a_{2k}$) is at most $\mathrm{\mid }A{\mathrm{\mid }}^{{\log }_2(2^k+2)}$, and ${\log }_2(2^k+2)$ is the best possible exponent. We also show that if $d\ge 0$ and $2\le k\le 10$, for any subset $A$ of a discrete cube $\{0,1{\}}^d$, the $k-$additive energy of $A$ (the number of $2k-$tuples $(a_1,a_2,\dots ,a_{2k})$ in $A^{2k}$ with $a_1+a_2+\cdots +a_k=a_{k+1}+a_{k+2}+\cdots +a_{2k}$) is at most $|A|^{\log_2{ \binom{2k}{k}}}$,and$\log_2{ \binom{2k}{k}}$ is the best possible exponent. We discuss the analogous problems for the sets $\{0,1,\dots ,n{\}}^d$ for $n\geq 2$.