Additive energies of subsets of discrete cubes

TL;DR

This paper investigates the asymptotic behavior of the smallest exponent controlling the additive energy of subsets in discrete cubes, establishing new bounds that refine previous trivial estimates for large dimensions.

Contribution

The paper provides non-trivial bounds on the exponent $t_n$ that governs additive energy in discrete cubes, improving understanding of its asymptotic behavior as $n$ grows.

Findings

Bounds on $t_n$ for large $n$ are established.

The lower bound approaches 3 minus a logarithmic term involving $rac{3 ext{sqrt}(3)}{4}$.

The upper bound approaches 3 minus a constant logarithmic term.

Abstract

For a positive integer $n \geq 2$, define $t_n$ to be the smallest number such that the additive energy $E(A)$ of any subset $A \subset \{0,1,\cdots,n-1\}^d$ and any $d$ is at most $|A|^{t_n}$. Trivially we have $t_n \leq 3$ and $$ t_n \geq 3 - \log_n\frac{3n^3}{2n^3+n} $$ by considering $A = \{0,1,\cdots,n-1\}^d$. In this note, we investigate the behavior of $t_n$ for large $n$ and obtain the following non-trivial bounds: $$ 3 - (1+o_{n\rightarrow\infty}(1)) \log_n \frac{3\sqrt{3}}{4} \leq t_n \leq 3 - \log_n(1+c), $$ where $c>0$ is an absolute constant.