Cancellation in additively twisted sums on $\mathrm{GL}(2)$ with non-linear phase

Abstract

Let ${\lambda }_g(n)$ be the Fourier coefficients of a holomorphic cusp modular form $g$ for ${\mathrm{SL}}_2(\mathbb{Z})$. The aim of this article is to get non-trivial bound on non-linearly additively twisted sums of the Fourier coefficients ${\lambda }_g(n)$. Precisely, we prove for any $3/4<\beta <3/2$, $\beta \ne 1$, the following non-trivial estimate $$\sum _{n\le N}{\lambda }_g(n)e(\alpha n^{\beta }){\ll }_{g,\alpha ,\beta ,\epsilon }{N}^{\frac{1}{2}+\frac{\beta }{3}+\epsilon }+N^{\frac{3}{2}-\frac{2\beta }{3}+\epsilon },$$ for any $\epsilon >0$. This is the first time that non-trivial estimate for such sums is achieved for $1<\beta <3/2$, breaking the barrier $\beta =1$ in the work of X. Ren and Y. Ye. It also improves their estimate in the range $9/10<\beta <1$. The key of our approach is a newly developed Bessel $\delta$-method.

This text has been merged into arXiv:1906.05485