Shifted Convolution Sums of ${A_f}(1,n)$ Cusp Forms with θ-Series

Abstract Let $f$ be the normalized Fourier coefficients of a Hecke–Maass cusp form $S{L_3}({\BBZ})$ for $${r_3}(n) = \# \left\{ {({n_1},{n_2},{n_3}) \in {\BBZ}{^3}:n_1^2 + n_2^2 + n_3^2 = n} \right\}.$$ and $1 \le h \le X$ Let $\phi (x)$ and $[1/2,1]$ be a smooth function compactly supported on $\varepsilon \gt 0$. It is proved that for any $$\mathop \sum \limits_{n \ge 1} {A_f}(1,n + h){r_3}(n)\phi \left( {{n \over X}} \right){ \ll _{f,\varepsilon }}{X^{{3 \over 2} - {1 \over 8} + \varepsilon }}$$, $h$ uniformly with respect to the shift $L = d_x^4 + u(x)$.