An Explicit Sum-Product Estimate in Fp

Abstract

Let 𝔽<inf><it>p</it></inf> be the field of residue classes modulo a prime number <it>p</it> and let <it>A</it> be a nonempty subset of 𝔽<inf><it>p</it></inf>. In this paper we give an explicit version of the sum-product estimate of Bourgain, Katz and Tao and Bourgain, Glibichuk and Konyagin on the size of max{|<it>A</it> + <it>A</it>|, |<it>AA</it>|}. In particular, our result implies that if 1 < |<it>A</it>| ≤ <it>p</it>7/13(log <it>p</it>)−4/13, then<fd>$$\\mathrm{max}\\{|A+A|\\hbox{ , }\\phantom{\\rule{0.4em}{0ex}}|\\mathrm{AA}|\\}\\gg \\frac{{|A|}^{\\mathrm{15/14}}}{{(\\mathrm{log}\\phantom{\\rule{0.4em}{0ex}}|A|)}^{\\mathrm{2/7}}}\\hbox{ . }$$</fd>