A slight improvement to Garaev's sum product estimate

Abstract

max(\A + A\,\AA\)>Ce\A\?-*. In the finite field setting this situation is much more complicated because the main tool, the Szemer?di-Trotter incidence theorem, does not hold in the same generality. It is known, via the work in [BKT], that if A is a subset of Fp, the field of p elements with p prime, and if p6 0, then one has the sum product estimate max(|? + A|,|AA|)>|,4|1+ for some e > 0. This result has found many applications in combinatorial problems and exponential sum estimates (see e.g. [BKT], [BGK], [G2]). Recently, Garaev [Gl] showed that when \A < pz, one has the estimate