Bounds on bilinear forms with Kloosterman sums

Bryce Kerr, Igor E. Shparlinski, Xiaosheng Wu, Ping Xi

Published 2022-04-11Version 1

We prove new bounds on bilinear forms with Kloosterman sums, complementing and improving a series of results by \'E. Fouvry, E. Kowalski and Ph. Michel (2014), V. Blomer, \'E. Fouvry, E. Kowalski, Ph. Michel and D. Mili\'cevi\'c (2017), E. Kowalski, Ph. Michel and W. Sawin (2019, 2020) and I. E. Shparlinski (2019). These improvements rely on new estimates for Type II bilinear forms with incomplete Kloosterman sums. We also establish new estimates for bilinear forms with one variable from an arbitrary set by introducing techniques from additive combinatorics over prime fields. As a typical application, an estimate for higher moments of averages of Kloosterman sums is also given.