A quantified version of Bourgain's sum-product estimate in \(\mathbb F_{p}\) for subsets of incomparable sizes (Q1010777)

Summary: Let \({\mathbb F}_p\) be the field of residue classes modulo a prime number \(p\). In this paper we prove that if \(A,B\subset {\mathbb F}_p^*,\) then for any fixed \(\varepsilon>0,\) \[ |A+A|+|AB|\gg \left(\min \left\{|B|, \frac{p}{|A|}\right\}\right)^{1/25-\varepsilon}|A|. \] This quantifies \textit{J. Bourgain}'s recent sum-product estimate [Int. J. Number Theory 1, No. 1, 1--32 (2005; Zbl 1173.11310)].