Bounded Gaps Between Primes in Multidimensional Hecke Equidistribution Problems

Using Duke’s large sieve inequality for Hecke Gr{"o}ssencharaktere and the new sieve methods of Maynard and Tao, we prove a general result on gaps between primes in the context of multidimensional Hecke equidistribution. As an application, for any fixed $0<ϵ<\frac{1}{2}$, we prove the existence of infinitely many bounded gaps between primes of the form $p=a^2+b^2$ such that $\mathrm{\mid }a\mathrm{\mid }<ϵ\sqrt{p}$. Furthermore, for certain diagonal curves $\mathcal{C}:a{x}^{\alpha }+b{y}^{\beta }=c$, we obtain infinitely many bounded gaps between the primes $p$ such that $\mathrm{\mid }p+1-\mathrm{\#}\mathcal{C}({\mathbb{F}}_p)\mathrm{\mid }<ϵ\sqrt{p}$.