Nonlinear twists of Rankin-Selberg coefficients in the eigenvalue aspect

Let $g$ be a fixed primitive Hecke–Maass cusp form, and let $f$ vary with spectral parameter $k_f=K$, at a fixed level and with trivial nebentypus. We study smooth sums of ${\lambda }_f(n){\lambda }_g(n)e(\alpha n^{\beta })$ for fixed real $\alpha \mathrm{\ne }0$. The crucial input is the smooth spectral second moment of Lau, Liu, and Ye, for which the window length can be taken to be $L=K^{1\mathrm{/}3+a}$ for arbitrarily small fixed $a>0$. We show that, for every $\kappa >0$, there is ${\beta }_0>0$ such that the sum is $O(K^{2\mathrm{/}3+\kappa }{X}^{1\mathrm{/}2})$ for $1\le X\le K^2$ and $0<\beta \le {\beta }_0$. We give a detailed Baire category argument which makes the dependence on the smooth weight uniform through one finite derivative norm. A second argument keeps the phase explicit, applies Voronoi summation and shifted convolution estimates, and gives the numerical bound $O(K^{423\mathrm{/}224+(5\mathrm{/}7)\beta +\epsilon })$ at $X=K^2$ for $0<\beta <5\mathrm{/}32$.