Are there arbitrarily long arithmetic progressions in the sequence of twin primes?

TL;DR

This paper explores the conditions under which twin primes can form arbitrarily long arithmetic progressions, assuming certain distribution hypotheses about primes, and provides bounds on the common difference.

Contribution

It establishes that under the assumption of a prime distribution level exceeding 1/2, arbitrarily long twin prime progressions exist with bounded differences, refining previous understanding.

Findings

Existence of arbitrarily long twin prime progressions under certain distribution assumptions

Bound on the difference d for progressions when the distribution level exceeds 0.971

Progression length and difference bounds depend on the prime distribution level

Abstract

The main result of the paper is that assuming that the level $\theta$ of distribution of primes exceeds 1/2, then there exists a positive $d\leq C(\theta)$ such that there are arbitrarily long arithmetic progressions with the property that $p'=p+d$ is the next prime for each element of the progression. If $\theta>0.971$, then the above holds for some $d\leq 16$.