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$\operatorname{GL}_{3}$-AUTOMORPHIC FORMS
We prove that sums of length about 
$q^{3/2}$ of Hecke eigenvalues of automorphic forms on 
$\operatorname{SL}_{3}(\mathbf{Z})$ do not correlate with 
$q$-periodic functions with bounded Fourier transform. This generalizes the earlier results of Munshi and Holowinsky–Nelson, corresponding to multiplicative Dirichlet characters, and applies, in particular, to trace functions of small conductor modulo primes.
We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums 
$\text{Kl}_{p}(a)$ , as 
$a$ varies over 
$\mathbf{F}_{p}^{\times }$ and as 
$p$ tends to infinity. Using independence of Kloosterman sheaves, we prove convergence in the sense of finite distributions to a specific random Fourier series. We also consider Birch sums, for which we can establish convergence in law in the space of continuous functions. We then derive some applications.
