On squares of Fourier coefficients twist exponential functions with applications

Let f be a Hecke-Maass cusp form of weight zero for SL2(Z) and lambda(f)(n) be the nth Fourier coefficient. For almost all v, we have [GRAPHICS] . 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(0.8297). For all alpha > 1 of type tau < infinity, we also show that [GRAPHICS] . where B alpha,ss := {[ alpha n+ ss]}(infinity) (n=1). This result relies heavily on a generalized double sum (see Theorem 1.3).