Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order

Abstract

Our first result is a ‘sum–product’ theorem for subsets A of the finite field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>𝔽</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> , p prime, providing a lower bound on max(| A + A |,| A · A |). As corollary, the second and main result provides new bounds on exponential sums associated to subgroups of the multiplicative group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>𝔽</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> <mml:mo>*</mml:mo> </mml:msubsup> </mml:math> .