Higher additive energies on discrete cubes

Source abstract

Let $m,n\geq2$ be integers. We study the least exponent $t_{m,n}$ such that the $m$-fold additive energy of any subset $A$ of the discrete cube $\{0,1,\cdots,n-1\}^d$ in any dimension $d$ satisfies $E_m(A)\leq |A|^{t_{m,n}}$. For every fixed $m$, we obtain the asymptotic formula $$t_{m,n}=2m-1-(1+o(1))\log_n \frac{(2m-1)^{m-1/2}}{(2m)^{m-1}} \qquad(n\to\infty).$$ For $m=2$, this gives $$t_{2,n} = 3 - (1+o(1)) \log_n\frac{3\sqrt{3}}{4},$$ proving a conjecture of the first author.