Analogues of the Balog–Wooley Decomposition for Subsets of Finite Fields and Character Sums with Convolutions
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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 a, b, c running through three arbitrary subsets of a finite field.
Keywords
Finite fields Convolution Inversions Sumsets Energy Character sumsMathematics Subject Classification
11B30 11T30Notes
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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