On the Discretized Sum-Product Problem

Creators: Guth, Larry; Katz, Nets Hawk; Zahl, Joshua

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Abstract

We give a new proof of the discretized ring theorem for sets of real numbers. As a special case, we show that if A ⊂ R is a (δ,1/2)₁-set in the sense of Katz and Tao, then either A+A or A.A must have measure at least |A|1−1/68⁠.

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© The Author(s) 2020. Published by Oxford University Press. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model). Received: 24 April 2018; Revision received: 31 October 2019; Accepted: 28 November 2019; Published: 13 January 2020. The authors would like to thank Brendan Murphy, Victor Lie, and Jianan Li for comments and corrections to a previous draft of this manuscript. The authors would also like to thank the anonymous referees for corrections and suggestions. This work was supported by a Simons Investigator Award [to L.G.]; and a NSERC Discovery Grant [to J.Z.].

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On the Discretized Sum-Product Problem

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