Monotone chains of Fourier coefficients of Hecke cusp forms

We prove general equidistribution statements (both conditional and unconditional) relating to the Fourier coefficients of arithmetically normalized holomorphic Hecke cusp forms $f_1,\ldots,f_k$ without complex multiplication, of equal weight, (possibly different) squarefree level and trivial nebentypus. As a first application, we show that for the Ramanujan $\tau$ function and any admissible $k$-tuple of distinct non-negative integers $a_1,\ldots,a_k$ the set $$\{n \in \mathbb{N} : |\tau(n+a_1)| < \cdots < |\tau(n+a_k)|\}$$ has positive natural density. This result improves upon recent work of Bilu, Deshouillers, Gun and Luca [Compos. Math. (2018), no. 11, 2441-2461]. Secondly, we make progress towards understanding the signed version by showing that $$\{n \in \mathbb{N} : \tau(n+a_1) < \tau(n+a_2) < \tau(n+a_3)\}$$ has positive relative upper density at least $1/6$ for any admissible triple of distinct non-negative integers $(a_1,a_2,a_3).$ More generally, for such chains of inequalities of length $k > 3$ we show that under the assumption of Elliott's conjecture on correlations of multiplicative functions, the relative natural density of this set is $1/k!.$ Previously results of such type were known for $k\le 2$ as consequences of works by Serre and by Matom\"{a}ki and Radziwill. Our results rely crucially on several key ingredients: i) a multivariate Erd\H{o}s-Kac type theorem for the function $n \mapsto \log|\tau(n)|$, conditioned on $n$ belonging to the set of non-vanishing of $\tau$, generalizing work of Luca, Radziwill and Shparlinski; ii) the recent breakthrough of Newton and Thorne on the functoriality of symmetric power $L$-functions for $\text{GL}(n)$ for all $n \geq 2$ and its application to quantitative forms of the Sato-Tate conjecture; and iii) the work of Tao and Ter\"{a}v\"{a}inen on the logarithmic Elliott conjecture.