Bounded Gaps Between Primes in Multidimensional Hecke Equidistribution Problems

TL;DR

This paper extends prime gap results to multidimensional Hecke equidistribution, proving infinitely many bounded gaps between primes of special forms and on certain algebraic curves using advanced sieve techniques.

Contribution

It introduces a general framework for prime gaps in multidimensional Hecke settings and applies it to primes of specific algebraic forms and curves.

Findings

Infinitely many bounded gaps between primes of the form p=a^2+b^2 with |a|<ε√p

Bounded gaps between primes related to diagonal curves over finite fields

Utilizes Duke's large sieve inequality and Maynard-Tao sieve methods

Abstract

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 $0<\epsilon<\frac{1}{2}$, 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 $\mathcal{C}:ax^{\alpha}+by^{\beta}=c$, we obtain infinitely many bounded gaps between the primes $p$ such that $|p+1-\#\mathcal{C}(\mathbb{F}_p)|<\epsilon\sqrt{p}$.