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

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 $3/4 < \beta < 3/2$, $\beta \neq 1$, the following non-trivial estimate $$\sum_{n \leq N}\lambda_g(n)\,e(\alpha\, n^{\beta})\ll_{g, \alpha, \beta, \varepsilon} N^{\frac{1}{2}+ \frac{\beta}{3} +\varepsilon} + N^{\frac{3}{2}-\frac {2\beta}{3} + \varepsilon},$$ for any $\varepsilon > 0$. 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.