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On higher energy decompositions and the sum–product phenomenon

Published online by Cambridge University Press:  03 July 2018

GEORGE SHAKAN
GEORGE SHAKAN*
Affiliation:
Department of Mathematics, University of Illinois, Urbana–Champaign e-mail: shakan2@illinois.edu
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Abstract

Let 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*} $

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 3 , November 2019 , pp. 599 - 617
Copyright
Copyright © Cambridge Philosophical Society 2018 

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