Bounds on an exponential sum arising in Boolean circuit complexity

Abstract

We study exponential sums of the form S = 2 − n ∑ x ∈ { 0 , 1 } n e m ( h ( x ) ) e q ( p ( x ) ) , where m , q ∈ Z + are relatively prime, p is a polynomial with coefficients in Z q , and h ( x ) = a ( x 1 + ⋯ + x n ) for some 1 ⩽ a < m . We prove an upper bound of the form 2 − Ω ( n ) on | S | . This generalizes a result of J. Bourgain, who establishes this bound in the case where q is odd. This bound has consequences in Boolean circuit complexity.