Non-linear additive twist of Fourier coefficients of $GL(3) \times GL(2)$ and $GL(3)$ Maass forms

Abstract: Let $\lambda_{\pi}(m,n)$ be the Fourier coefficients of a Hecke-Maass cusp form $\pi$ for $SL(3,\mathbb{Z})$ and $\lambda_{f}(n)$ be the Fourier coefficients of Hecke-eigen form $f$ for $SL(2,\mathbb{Z})$. The aim of this article is to get a non-trivial bound on the sum which is non-linear additive twist of the coefficients $\lambda_{\pi}(m,n)$ and $\lambda_{f}(n)$. More precisely, for any $0 < \beta < 1$ and $\epsilon>0$, we have $$\sum_{n=1}^{\infty} \lambda_{\pi}(r,n) \, e\left(\alpha n^{\beta}\right) V\left(\frac{n}{X}\right) \ll_{\pi,\epsilon} \alpha \sqrt{\beta}r^{\frac{7}{6}}X^{\frac{3}{4}+\frac{9\beta}{28}+ \epsilon}.$$ and $$\sum_{n=1}^{\infty} \lambda_{\pi}(r,n) \, \lambda_{f}(n) \, e\left(\alpha n^{\beta}\right) V\left(\frac{n}{X}\right) \ll_{\pi, f,\epsilon} (\alpha \beta)^{\frac{3}{2}} rX^{\frac{3}{4}+\frac{29\beta}{44}+\epsilon},$$ where $V(x)$ is a smooth function supported in $[1,2]$ and satisfying $V^{(j)}(x) \ll_{j} 1$.