Patterns of primes in joint Sato--Tate distributions

Abstract

For $j=1,2$, let $f_j(z) = \sum_{n=1}^{\infty} a_{j}(n) e^{2\pi i nz}$ be a holomorphic, non-CM cuspidal newform of even weight $k_j \ge 2$ with trivial nebentypus. For each prime $p$, let $\theta_{j}(p)\in[0,\pi]$ be the angle such that $a_j(p) = 2p^{(k-1)/2} \cos \theta_{j}(p)$. The now-proven Sato--Tate conjecture states that the angles $(\theta_j(p))$ equidistribute with respect to the measure $d\mu_{\mathrm ST} = \frac{2}{\pi}\sin^2\theta\,d\theta$. We show that, if $f_1$ is not a character twist of $f_2$, then for subintervals $I_1,I_2 \subset [0,\pi]$, there exist infinitely many bounded gaps between the primes $p$ such that $\theta_1(p) \in I_1$ and $\theta_2(p) \in I_2$. We also prove a common generalization of the bounded gaps with the Green--Tao theorem.