Functiones et Approximatio Commentarii Mathematici

Counting additive decompositions of quadratic residues in finite fields

Simon R. Blackburn, Sergei V. Konyagin, and Igor E. Shparlinski

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We say that a set $\mathcal{S}$ is additively decomposed into two sets $\mathcal{A}$ and $\mathcal{B}$ if $\mathcal{S} = \{a+b: a\in \mathcal{A}, b \in \mathcal{B}\}$. A. Sárközy has recently conjectured that the set $\mathcal{Q}$ of quadratic residues modulo a prime $p$ does not have nontrivial decompositions. Although various partial results towards this conjecture have been obtained, it is still open. Here we obtain a nontrivial upper bound on the number of such decompositions.

Article information

Funct. Approx. Comment. Math., Volume 52, Number 2 (2015), 223-227.

First available in Project Euclid: 18 June 2015

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11B13: Additive bases, including sumsets [See also 05B10]
Secondary: 11L40: Estimates on character sums

additive decompositions finite fields quadratic nonresidues character sums


Blackburn, Simon R.; Konyagin, Sergei V.; Shparlinski, Igor E. Counting additive decompositions of quadratic residues in finite fields. Funct. Approx. Comment. Math. 52 (2015), no. 2, 223--227. doi:10.7169/facm/2015.52.2.3.

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