Beyond the Weyl barrier for GL(2) exponential sums

Abstract

In this paper, we use the Bessel δ-method, along with new variants of the van der Corput method in two dimensions, to prove non-trivial bounds for GL(2) exponential sums beyond the Weyl barrier. More explicitly, for sums of GL(2) Fourier coefficients twisted by e(f(n)), with length N and phase f(n)=Nβlog⁡n/2π or anβ, non-trivial bounds are established for β<1.63651..., which is beyond the Weyl barrier at β=3/2.