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

arXiv comment26 pages

References in corpus (3)

• On sums of Szemerédi--Trotter sets
• New results on sum-products in R
• Some remarks on the Balog-Wooley decomposition theorem and quantities D^+, D^\times

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