On the sum-product phenomenon in arbitrary finite fields and its applications

Abstract

This thesis establishes new quantitative results in several problems relating to the sum-product phenomenon in arbitrary finite fields. We give new estimates of exponential sums, two-variable expanders and point-line incidences. We also consider an energy variant of the sum-product problem, extending a result of Balog and Wooley to arbitrary finite fields. Our approach towards the sum-product problem relies on the so-called additive pivot technique and our results hold under certain structural assumptions, requiring that a set is not largely contained in a proper subfield. This is in contrast to other recent developments, which rely on a result on point-plane incidences due to Rudnev and hold only for sets which are bounded in size in terms of the characteristic of the field.