On higher energy decompositions and the sum–product phenomenon

Abstract

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