On shifted convolution sums of $\mathrm{GL}(3)$-Fourier coefficients with an average over shifts

Abstract

Let $F$ be a Hecke-Maass cusp form for ${\mathrm{SL}}_3(\mathbb{Z})$ and $A(m,n)$ be its normalized Fourier coefficients. Let $V$ be a smooth function, compactly supported on $[1,2]$ and satisfying $V(y{)}^j{\ll }_j{y}^{-j}$ for any $j\in \mathbb{N}\cup \{0\}$. In this article we prove a power-saving upper bound for the `average' shifted convolution sum \begin{equation*} \sum_{h}\sum_{n}A(1,n)A(1,n+h)V\left(\frac{n}{N}\right)V\left(\frac{h}{H}\right), \end{equation*} for the range $N^{1/2-\epsilon }\ge H\ge N^{1/6+\epsilon }$, for any $\epsilon >0$. This is an improvement over the previously known range $N^{1/2-\epsilon }\ge H\ge N^{1/4+\epsilon }$.

33 pages