On the asymptotics of the shifted sums of Hecke eigenvalue squares

Abstract: The purpose of this paper is to obtain asymptotics of shifted sums of Hecke eigenvalue squares on average. We show that for $X^{\frac{2}{3}+\epsilon} < H <X^{1-\epsilon},$ there are constants $B_{h}$ such that $$ \sum_{X\leq n \leq 2X} \lambda_{f}(n)^{2}\lambda_{f}(n+h)^{2}-B_{h}X=O_{f,A,\epsilon}\big(X (\log X)^{-A}\big)$$ for all but $O_{f,A,\epsilon}\big(H(\log X)^{-3A}\big)$ integers $h \in [1,H]$ where ${\lambda_{f}(n)}{n\geq1}$ are normalized Hecke eigenvalues of a fixed holomorphic cusp form $f.$ Our method is based on the Hardy-Littlewood circle method. We divide the minor arcs into two parts $m{1}$ and $m_{2}.$ In order to treat $m_{2},$ we use the Hecke relations, a bound of Miller to apply some arguments from a paper of Matom\"{a}ki, Radziwill and Tao. We apply Parseval's identity and Gallagher's lemma so as to treat $m_{1}.$