Exponential sums of squares of Fourier coefficients of cusp forms

Abstract

We prove nontrivial estimates for linear sums of squares of Fourier coefficients of holomorphic and Maass cusp forms twisted by additive characters. For holomorphic forms f, we show that if $$|\alpha -a/q| \le 1/q^2$$ with $$(a,q)=1$$ , then for any $$\varepsilon >0$$ , $$\begin{aligned} \qquad \qquad \sum _{n\leqslant X}{\lambda _f(n)}^2 e(n\alpha ) \ll _{f, \varepsilon } X^{{\frac{4}{5}}+\varepsilon } \quad \text {for} \ X^{{\frac{1}{5}}} \ll q \ll X^{{\frac{4}{5}}}. \end{aligned}$$ Moreover, for any $$\varepsilon > 0,$$ there exists a set $$S \subset (0, 1)$$ with $$\mu (S)=1$$ such that for every $$\alpha \in S$$ , there exists $$X_0=X_0(\alpha )$$ such that the above inequality holds true for any $$\alpha \in S$$ and $$X \geqslant X_0(\alpha ).$$ A weaker bound for Maass cusp forms is also established.