On common energies and sumsets II

TL;DR

The paper investigates how the size of a sumset $|A+A|$ ties to the minimal common energy over disjoint partitions of a finite set $A$ in an abelian group. It establishes a subexponential bound: if every disjoint split $A=X\cup Y$ satisfies $E(X,Y) \ge \frac{|X|^2|Y|^2}{K|A|}$, then $\frac{|A+A|}{|A|}$ is bounded by $\exp(O(K^{1/3}\log^2 K))$, and it proves near-tightness up to exponent gaps via a constructed lower bound. The core argument weaves energy methods, a Balog–Szemerédi–Gowers step, and a greedy iterative scheme to relate disjoint-partition energy to doubling, culminating in the main bound $\mathcal{D}[A] \le \exp(O(\mathcal{E}_*[A]^{1/3}\log^2 \mathcal{E}_*[A]))$. An explicit counterexample demonstrates subexponential tightness, and the results yield a direct, parameter-optimized proof of the arithmetic regularity lemma of Green–Sisask, highlighting improved dependence. Collectively, the work advances understanding of how energy constraints govern sumset growth and provides refined structure theorems in additive combinatorics.

Abstract

We continue to study the relationship between the size of the sum of a set and the common energy of its subsets. We find a rather sharp subexponential dependence between the doubling constant of a set $A$ and the minimal common energy taken over all partitions of $A$ into two disjoint subsets. As an application, we give a proof of the well--known arithmetic regularity lemma with better dependence on parameters.

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Proposition 5
• proof
• Theorem 6
• proof
• Theorem 7