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

26 pages