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An update on the sum-product problem

Part of: Sequences and sets

Published online by Cambridge University Press:  11 October 2021

MISHA RUDNEV  and
SOPHIE STEVENS
MISHA RUDNEV
Affiliation:
School of Mathematics, Fry Building, University of Bristol, Bristol, BS8, 1UG e-mail: misha.rudnev@bristol.ac.uk
SOPHIE STEVENS
Affiliation:
Johann Radon Institut, (RICAM), Altenberger Strasse 69, 4040 Linz, Austria e-mail: sophie.stevens@ricam.oeaw.ac.at
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Abstract

We improve the best known sum-product estimates over the reals. We prove that

\[\max(|A+A|,|A+A|)\geq |A|^{\frac{4}{3} + \frac{2}{1167} - o(1)}\,,\]
for a finite $A\subset \mathbb {R}$ , following a streamlining of the arguments of Solymosi, Konyagin and Shkredov. We include several new observations to our techniques.

Furthermore,

\[|AA+AA|\geq |A|^{\frac{127}{80} - o(1)}\,.\]
Besides, for a convex set A we show that
\[|A+A|\geq |A|^{\frac{30}{19}-o(1)}\,.\]
This paper is largely self-contained.

MSC classification

Primary: 11B13: Additive bases, including sumsets
Secondary: 11B30: Arithmetic combinatorics; higher degree uniformity 11B75: Other combinatorial number theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 2 , September 2022 , pp. 411 - 430
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

†

Partially supported by the Leverhulme Trust Grant RPG–2017–371.

‡

Supported by the Austrian Science Fund FWF Project P 30405-N32.

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