Bounded Gaps Between Primes in Multidimensional Hecke Equidistribution Problems

Abstract

Using Duke's large sieve inequality for Hecke Gr{ö}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 $0<\epsilon <\frac{1}{2}$, we prove the existence of infinitely many bounded gaps between primes of the form $p=a^2+b^2$ such that $|a|<\epsilon \sqrt{p}$. 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 $|p+1-\#\mathcal{C}({\mathbb{F}}_p)|<\epsilon \sqrt{p}$.