On $\rm GL_3$ Fourier coefficients over values of mixed powers
Yanxue Yu
TL;DR
The paper bounds GL$_3$ Fourier coefficients $A_(n,1)$ on sparse mixed-power values, extending circle-method and Voronoi techniques to the rank-3 setting. Using GL$_3$ Voronoi summation, Gauss sums, and Langlands-parameter bounds, it develops a detailed major/minor arc analysis for sums of the form $\sum A_(n,1)\omega(n/X)$ over $n=n_1^r+\cdots+n_\ell^r+n_{\ell+1}^s$ with $r,s\ge2$ and $\ell\ge2^{r-1}$, yielding nontrivial upper bounds. The main result, Theorem 1.1, provides a piecewise bound depending on $\ell/r+1/s$ relative to $7/2$, with explicit $\theta_0=\min\{1/r,1/s\}$-dependent terms, and a secondary bound when $\ell/r+1/s<7/2$. This advances understanding of GL$_3$ coefficient distributions on sparse sequences and showcases the power of combining circle method with high-rank Voronoi theory for automorphic forms.
Abstract
Let $A_π(n,1)$ be the $(n,1)$-th Fourier coefficient of the Hecke-Maass cusp form $π$ for $\rm SL_3(\mathbb{Z})$ and $ ω(x)$ be a smooth compactly supported function. In this paper, we prove a nontrivial upper bound for the sum $$\sum_{n_1,\cdots,n_\ell,n_{\ell+1}\in\mathbb{Z}^+ \atop n=n_1^r+\cdots+n_{\ell}^r+n_{\ell+1}^s} A_π(n,1)ω\left(n/X\right),$$ where $r\geq2$, $s\geq 2$ and $\ell\geq 2^{r-1}$ are integers.