Sums of products of Kloosterman sums to prime power moduli

Abstract

Abstract We prove new bounds on complete sums of products of $$k$$ k additively shifted Kloosterman sums to odd high prime power moduli $$q=p^n$$ q = p n , which feature substantially stronger power savings (about $$q^{-1/\lceil k/2\rceil}$$ q - 1 / ⌈ k / 2 ⌉ in generic configurations) and a novel quantification of the alignment among the shifts. We prove our bounds by developing a method to estimate complete exponential sums with a broad class of phases well-controlled by a specified number of initial terms in a property reflecting p -adic differentiability.