Resonances and $\Omega$-results for Exponential Sums Related to Maass Forms for $\mathrm{SL}(n,\mathbb{Z})$

Abstract

We obtain resonances for short exponential sums involving Fourier coefficients of Maass forms for $\mathrm{SL}(n,\mathbb{Z})$. This involves deriving asymptotics for the integrals appearing in the $\mathrm{GL}(n)$ Voronoi summation formula. As an application, we also prove an $\Omega$-result for short sums of Fourier coefficients.