Shifted Convolution Sums for $GL(3)\times GL(2)$ Averaged over weighted sets

TL;DR

This paper establishes bounds for shifted convolution sums involving $GL(3)$ and $GL(2)$ Fourier coefficients averaged over weighted sets, improving previous results and connecting to the structure of factorizable moduli.

Contribution

It provides new bounds for averaged shifted sums with arbitrary weights and links the structure of these sums to the factorizable moduli in Jutila's circle method.

Findings

Improved bounds for shifted convolution sums with weighted averages.

Generalization of Sun's bound to arbitrary weights.

Connection between shifted sums and factorizable moduli structure.

Abstract

Let $A(1,m)$ be the Fourier coefficients of a $SL(3,\mathbb{Z})$ Hecke-Maass cusp form $\pi_1$ and $\lambda(m)$ be those of a $SL(2,\mathbb{Z})$ Hecke holomorphic or Hecke-Mass cusp form $\pi_2$. Let $H\subset[\![ -X^{1-\varepsilon},X^{1+\varepsilon}]\!]$ and $\{a(h)\}_{h\in H}\subset\mathbb{C}$ be a sequence. We show that if $H\subset \ell+[\![ 0,X^{1/2+\varepsilon}]\!] $ for some $\ell\geq0$, \begin{align*} D_{a,H}(X):=\frac{1}{|H|}\sum_{h\in H}a(h)\sum_{m=1}^\infty A(1,m)\lambda(rm+h)V\left(\frac{m}{X}\right)\ll_{\pi_1,\pi_2,\varepsilon} \frac{X^{1+\varepsilon}}{|H|}\|a\|_2 \end{align*} for any $\varepsilon>0$, and a similar bound when $|H|$ is big. This improves Sun's bound and generalizes it to an average with arbitrary weights. Moreover, we demonstrate how one can recover the factorizable moduli structure given by the Jutila's circle method via studying a shifted sum with weighted average. This allows us to recover Munshi's bound on the shifted sum with a fixed shift without using the Jutila's circle method.