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A robust version of Freiman's 3k–4 Theorem and applications

Published online by Cambridge University Press:  27 March 2018

XUANCHENG SHAO  and
WENQIANG XU
XUANCHENG SHAO
Affiliation:
Department of Mathematics, 715 Patterson Office Tower, University of Kentucky, Lexington, KY, 40506, U.S.A. e-mail: xuancheng.shao@uky.edu
WENQIANG XU
Affiliation:
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT. e-mail: wenqiang.xu@ucl.ac.uk
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Abstract

We prove a robust version of Freiman's 3k – 4 theorem on the restricted sumset A+ΓB, which applies when the doubling constant is at most (3+$\sqrt{5}$)/2 in general and at most 3 in the special case when A = −B. As applications, we derive robust results with other types of assumptions on popular sums, and structure theorems for sets satisfying almost equalities in discrete and continuous versions of the Riesz–Sobolev inequality.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 166 , Issue 3 , May 2019 , pp. 567 - 581
Copyright
Copyright © Cambridge Philosophical Society 2018 

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Footnotes

†

Supported by a Glasstone Research Fellowship.

‡

Supported by a London Mathematics Society Undergraduate Research Bursary and the Mathematical Institute at University of Oxford.

References

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