Nonlinear twists of Rankin-Selberg coefficients in the eigenvalue aspect

Source abstract

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 $λ_f(n)λ_g(n)e(αn^β)$ for fixed real $α\ne0$. The crucial input is the smooth spectral second moment of Lau, Liu, and Ye, for which the window length can be taken to be $L=K^{1/3+a}$ for arbitrarily small fixed $a>0$. We show that, for every $κ>0$, there is $β_0>0$ such that the sum is $O(K^{2/3+κ}X^{1/2})$ for $1\le X\le K^2$ and $0<β\leβ_0$. 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 $O(K^{423/224+(5/7)β+\varepsilon})$ at $X=K^2$ for $0<β<5/32$.