Moments and oscillations of exponential sums related to cusp forms

Abstract: We consider large values of long linear exponential sums involving Fourier coefficients of holomorphic cusp forms. The sums we consider involve rational linear twists $e(nh/k)$ with sufficiently small denominators. We prove both pointwise upper bounds and bounds for the frequency of large values. In particular, the $k$-aspect is treated. As an application we obtain upper bounds for all the moments of the sums in question. We also give the asymptotics with the right main term for fourth moments. We also consider the mean square of very short sums, proving that on average short linear sums with rational additive twists exhibit square root cancellation. This result is also proved in a slightly sharper form. Finally, the consideration of moment estimates for both long and short exponential sums culminates in a result concerning the oscillation of the long linear sums. Essentially, this result says that for a positive proportion of time, such a sum stays in fairly long intervals, where its order of magnitude does not drop below the average order of magnitude and where its argument is in a given interval of length $3\pi/2+\varepsilon$ and so can not wind around the origin.