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Sumsets in quadratic residues

Volume 164 / 2014

I. D. Shkredov Acta Arithmetica 164 (2014), 221-243 MSC: 11B13, 11B50, 11B75. DOI: 10.4064/aa164-3-2


We describe all sets $A \subseteq {\mathbb F}_p$ which represent the quadratic residues $R \subseteq {\mathbb F}_p$ in the sense that $R=A+A$ or $R=A\mathbin {\hat{+} }A$. Also, we consider the case of an approximate equality $R \approx A+A$ and $R \approx A\mathbin {\hat{+}} A$ and prove that $A$ is then close to a perfect difference set.


  • I. D. ShkredovDivision of Algebra and Number Theory
    Steklov Mathematical Institute
    Gubkina St. 8
    Moscow, Russia 119991
    Delone Laboratory of Discrete and Computational Geometry
    Yaroslavl' State University
    Sovetskaya St. 14
    Yaroslavl', Russia 150000

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