Patterns of primes in joint Sato--Tate distributions

Abstract

For $j=1,2$, let $f_j(z) = \sum_{n=1}^{\infty} a_{j}(n) e^{2πi nz}$ be a holomorphic, non-CM cuspidal newform of even weight $k_j \ge 2$ with trivial nebentypus. For each prime $p$, let $θ_{j}(p)\in[0,π]$ be the angle such that $a_j(p) = 2p^{(k-1)/2} \cos θ_{j}(p)$. The now-proven Sato--Tate conjecture states that the angles $(θ_j(p))$ equidistribute with respect to the measure $dμ_{\mathrm ST} = \frac{2}π\sin^2θ\,dθ$. We show that, if $f_1$ is not a character twist of $f_2$, then for subintervals $I_1,I_2 \subset [0,π]$, there exist infinitely many bounded gaps between the primes $p$ such that $θ_1(p) \in I_1$ and $θ_2(p) \in I_2$. We also prove a common generalization of the bounded gaps with the Green--Tao theorem.