Additive energies on discrete cubes

TL;DR

This paper establishes optimal bounds on higher and additive energies of subsets within discrete cubes, revealing precise exponents and extending the results to larger discrete sets.

Contribution

It provides the first sharp bounds for higher and additive energies on discrete cubes, including the exact exponents and their optimality, and discusses extensions to larger discrete sets.

Findings

Upper bounds for higher energies with optimal exponents

Upper bounds for additive energies with optimal exponents

Extension discussions for larger discrete sets

Abstract

We prove that for $d\geq 0$ and $k\geq 2$, for any subset $A$ of a discrete cube $\{0,1\}^d$, the $k-$higher energy of $A$ (the number of $2k-$tuples $(a_1,a_2,\dots,a_{2k})$ in $A^{2k}$ with $a_1-a_2=a_3-a_4=\dots=a_{2k-1}-a_{2k}$) is at most $|A|^{\log_{2}(2^k+2)}$, and $\log_{2}(2^k+2)$ is the best possible exponent. We also show that if $d\geq 0$ and $2\leq k\leq 10$, for any subset $A$ of a discrete cube $\{0,1\}^d$, the $k-$additive energy of $A$ (the number of $2k-$tuples $(a_1,a_2,\dots,a_{2k})$ in $A^{2k}$ with $a_1+a_2+\dots+a_k=a_{k+1}+a_{k+2}+\dots+a_{2k}$) is at most $|A|^{\log_2{ \binom{2k}{k}}}$, and $\log_2{ \binom{2k}{k}}$ is the best possible exponent. We discuss the analogous problems for the sets $\{0,1,\dots,n\}^d$ for $n\geq 2$.