Exponential sums of squares of Fourier coefficients of cusp forms.

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 | α - a / q | ≤ 1 / q 2 with (a , q) = 1 , then for any ε > 0 , ∑ n ⩽ X λ f (n) 2 e (n α) ≪ f , ε X 4 5 + ε for X 1 5 ≪ q ≪ X 4 5. <graphic href="12044_2019_550_Article_Equ23.gif"></graphic> Moreover, for any ε > 0 , there exists a set S ⊂ (0 , 1) with μ (S) = 1 such that for every α ∈ S , there exists X 0 = X 0 (α) such that the above inequality holds true for any α ∈ S and X ⩾ X 0 (α). A weaker bound for Maass cusp forms is also established. [ABSTRACT FROM AUTHOR]