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Analogues of the Balog–Wooley Decomposition for Subsets of Finite Fields and Character Sums with Convolutions

  • Oliver Roche-Newton
  • Igor E. ShparlinskiEmail author
  • Arne Winterhof
Article
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Abstract

Balog and Wooley have recently proved that any subset \({\mathcal {A}}\) of either real numbers or of a prime finite field can be decomposed into two parts \({\mathcal {U}}\) and \({\mathcal {V}}\), one of small additive energy and the other of small multiplicative energy. In the case of arbitrary finite fields, we obtain an analogue that under some natural restrictions for a rational function f both the additive energies of \({\mathcal {U}}\) and \(f({\mathcal {V}})\) are small. Our method is based on bounds of character sums which leads to the restriction \(\# {\mathcal {A}}> q^{1/2}\), where q is the field size. The bound is optimal, up to logarithmic factors, when \(\# {\mathcal {A}}\ge q^{9/13}\). Using \(f(X)=X^{-1}\) we apply this result to estimate some triple additive and multiplicative character sums involving three sets with convolutions \(ab+ac+bc\) with variables abc running through three arbitrary subsets of a finite field.

Keywords

Finite fields Convolution Inversions Sumsets Energy Character sums 

Mathematics Subject Classification

11B30 11T30 

Notes

Acknowledgements

The authors thank Brendan Murphy, Misha Rudnev, Ilya Shkredov and Sophie Stevens for helpful conversations. Oliver Roche-Newton and Arne Winterhof were supported by the Austrian Science Fund FWF Projects F5509 and F5511-N26, respectively, which are part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications” as well as FWF project P30405-N32. Igor E. Shparlinski was supported by the Australian Research Council Grant DP170100786.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Oliver Roche-Newton
    • 1
  • Igor E. Shparlinski
    • 2
    Email author
  • Arne Winterhof
    • 1
  1. 1.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria
  2. 2.School of Mathematics and StatisticsUniversity of New South WalesKensingtonAustralia

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