On higher energy decompositions and the sum-product phenomenon

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 $$ |A+A| + |A A| \gtrsim |A|^{4/3 + 5/5277}.$$