Quantitative sum product estimates on different sets (Q1010700)

Let \(\mathbb F_p\) denote the finite field with \(p\) elements, where \(p\) is a prime. The paper shows that for every \(A,B\subset \mathbb F_p\), if \(|B|\leq |A| < p^{1/2}\), then \(\max(|A+B|, |AB|) {\geq \atop \approx} |B|^{14/18}|A|^{5/18}\). This improves on results of \textit{J. Bourgain, N. Katz} and \textit{T. Tao} [Geom. Funct. Anal. 14, No. 1, 27--57 (2004; Zbl 1145.11306)] and \textit{J. Bourgain, A. Glibichuk} and \textit{S. Konyagin} [J. Lond. Math. Soc., II. Ser. 73, No. 2, 380--398 (2006; Zbl 1093.11057)].