Orthogonality relation between $GL(3)$ coefficients and additive characters with linear and non-linear phases

Abstract

We study the distribution and oscillatory properties of prime-supported Fourier coefficients attached to a Hecke--Maass form for $\mathrm{SL}(3,\mathbb{Z})$ twisted by additive characters. For linear phases involving irrationals of finite type, we construct an automorphic variant of Vaughan's identity and exploit bilinear cancellation to obtain an unconditional power-saving estimate. For non-linear phases $e(\alpha n^{\theta })$ with $0<\theta <1$, we refine the Type~II bilinear forms via Weyl differencing and modern exponent pair methods. Combined with the analysis of untwisted $\mathrm{GL}(3)$ sums, this framework breaks the $2/3$-barrier of Jiang and Lü and widens the admissible range of non-trivial cancellation.