On Hecke eigenvalues of cusp forms in almost all short intervals

Abstract

Let $\psi$ be a function such that $\psi(x) \rightarrow \infty$ as $x \rightarrow \infty.$ Let $\lambda_{f}(n)$ be the $n$-th Hecke eigenvalue of a fixed holomorphic cusp form $f$ for $SL(2,\mathbb{Z}).$ We show that for any real valued function $h(x)$ such that $(\log X)^{2-2\alpha} \ll h(X) =o(X),$ $$\sum_{n=x}^{x+h(X)} |\lambda_{f}(n)| \ll_{f} h(X)\psi(X)(\log X)^{\alpha-1}$$ for all but $O_{f}( X\psi(X)^{-2})$ many integers $x\in [X,2X-h(X)],$ in which $\alpha$ is the average value of $|\lambda_{f}(p)|$ over primes. We generalize this for $|\lambda_{f}(n)|^{2^{k}}$ for $k \in \mathbb{Z^{+}}.$