Higher additive energies on discrete cubes

Xuancheng Shao; Yu-Chen Sun

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arXiv · 2610.07490

Higher additive energies on discrete cubes

Xuancheng Shao·Yu-Chen Sun

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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.

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Xuancheng Shao, Yu-Chen Sun. 2026-10-05. Higher additive energies on discrete cubes. https://arxiv.org/abs/2610.07490

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