On squares of Fourier coefficients twist exponential functions with applications.

Let f be a Hecke–Maass cusp form of weight zero for S L 2 (ℤ) and λ f (n) be the nth Fourier coefficient. For almost all , we have ∑ n ≤ x λ f 2 (n) e (n) ≪ x 0. 8 1 2 5 + , which improves the result of Acharya [Exponential sums of squares of Fourier coefficients of cusp forms, Proc. Indian Acad. Sci. Math. Sci. 130 (2020) 24], who showed an upper bound larger than x 0. 8 2 9 7. For all α > 1 of type τ < ∞ , we also show that ∑ 1 ≤ n ≤ x λ f 2 ([ α n + β ]) − α − 1 ∑ 1 ≤ n ≤ [ α x + β ] λ f 2 (n) ≪ x 1 3 / 1 6 + + x 1 − 3 / 4 (τ + 1) + , where ℬ α , β : = { [ α n + β ] } n = 1 ∞. This result relies heavily on a generalized double sum (see Theorem 1.3). [ABSTRACT FROM AUTHOR]