Query complexity and the polynomial Freiman–Ruzsa conjecture

We prove a query complexity variant of the weak polynomial Freiman–Ruzsa conjecture in the following form. For any ϵ>0, a set A⊂Zd with doubling K has a subset of size at least K−4ϵ|A| with coordinate query complexity at most ϵlog2⁡|A|. We apply this structural result to give a simple proof of th...

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Bibliographic Details
Published inAdvances in mathematics (New York. 1965) Vol. 392; p. 108043
Main Authors Zhelezov, Dmitrii, Pálvölgyi, Dömötör
Format Journal Article
LanguageEnglish
Published Elsevier Inc 03.12.2021
Subjects
Polynomial Freiman–Ruzsa
Sum-product
Sumsets
Sum-product
11B30
Polynomial Freiman–Ruzsa
Sumsets
Online AccessGet full text
ISSN0001-8708
1090-2082
1090-2082
DOI10.1016/j.aim.2021.108043

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Summary:We prove a query complexity variant of the weak polynomial Freiman–Ruzsa conjecture in the following form. For any ϵ>0, a set A⊂Zd with doubling K has a subset of size at least K−4ϵ|A| with coordinate query complexity at most ϵlog2⁡|A|. We apply this structural result to give a simple proof of the “few products, many sums” phenomenon for integer sets. The resulting bounds are explicit and improve on the seminal result of Bourgain and Chang.
ISSN:0001-8708
1090-2082
1090-2082
DOI:10.1016/j.aim.2021.108043