PreprintCC0arXiv
Higher additive energies on discrete cubes
Abstract
Let m,n≥2 be integers. We study the least exponent tm,n such that the m-fold additive energy of any subset A of the discrete cube {0,1,⋯,n−1}d in any dimension d satisfies Em(A)≤ |A|tm,n. For every fixed m, we obtain the asymptotic formula tm,n=2m−1−(1+o(1))logn ((2m−1)m−1/2)/((2m)m−1) (n→∞). For m=2, this gives t2,n = 3 − (1+o(1)) logn3√3/4, proving a conjecture of the first author.
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BibTeX
@misc{shao2026higher,
title = {{Higher additive energies on discrete cubes}},
author = {Xuancheng Shao and Yu-Chen Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07490},
eprint = {2610.07490},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07490},
note = {CC0},
}
RIS
TY - UNPB
TI - Higher additive energies on discrete cubes
AU - Xuancheng Shao
AU - Yu-Chen Sun
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07490
UR - https://arxiv.org/abs/2610.07490
AB - Let m,n≥2 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,⋯,n−1}^d in any dimension d satisfies E_m(A)≤ |A|^t_m,n. For every fixed m, we obtain the asymptotic formula t_m,n=2m−1−(1+o(1))log_n ((2m−1)^m−1/2)/((2m)^m−1) (n→∞). For m=2, this gives t_2,n = 3 − (1+o(1)) log_n3√3/4, proving a conjecture of the first author.
ER -