On higher energy decompositions and the sum-product phenomenon

TL;DR

This paper improves the Balog-Wooley decomposition for finite sets of real numbers, leading to stronger sum-product estimates by refining energy bounds and partitioning strategies.

Contribution

It provides a quantitatively improved decomposition of sets into parts with controlled additive and multiplicative energies, advancing sum-product estimate techniques.

Findings

Enhanced bounds on additive and multiplicative energies.

Improved sum-product inequality: |A+A| + |A A| rom previous bounds.

Refined decomposition methods for finite real sets.

Abstract

Let $A \subset \mathbb{R}$ be finite. We quantitatively improve the Balog-Wooley decomposition, that is $A$ can be partitioned into sets $B$ and $C$ such that $$\max\{E^+(B) , E^{\times}(C)\} \lesssim |A|^{3 - 7/26}, \ \ \max \{E^+(B,A) , E^{\times}(C, A) \}\lesssim |A|^{3 - 1/4}.$$ We use similar decompositions to improve upon various sum-product estimates. For instance, we show $$ |A+A| + |A A| \gtrsim |A|^{4/3 + 5/5277}.$$