Abstract
For a positive integer n ≥ 2, define tn to be the smallest number such that the additive energy E (A) of any subset A ⊂ {0, 1, · · · , n − 1}d and any d is at most |A|tn . Trivially, we have tn ≤ 3 and tn ≥ 3 − logn 3n3 2n3 + n by considering A = {0, 1, · · · , n − 1}d. In this note, we investigate the behaviour of tn for large n and obtain the following non-trivial bounds: 3 − (1 + on→∞(1)) logn 3√3 4 ≤ tn ≤ 3 − logn(1 + c), where c > 0 is an absolute constant.
Document Type
Article
Publication Date
2024
Digital Object Identifier (DOI)
https://doi.org/10.1017/prm.2024.126
Funding Information
X.S. was supported by NSF grant DMS-2200565. Thanks to Ali Alsetri for pointing out the reference [11] and to Andrew Granville for helpful discussions.
Repository Citation
Shao, Xuancheng, "Additive energies of subsets of discrete cubes" (2024). Mathematics Faculty Publications. 48.
https://uknowledge.uky.edu/math_facpub/48
Notes/Citation Information
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.