Equidistribution of the Fourier coefficients of Hecke cusp forms and applications

I will discuss joint work with A. Mangerel, where we establish a general
joint equidistribution result for the Fourier coefficients of Hecke cusp
forms. One simple to state consequence of such a result is that the set of
integers with, say, $\tau(n+2)<\tau(n+1)<\tau(n)$ where $\tau$ is
the Ramanujan $\tau$-function, has a positive upper density
(previously, even the infinitude of such a set was unknown).