On the ratio of consecutive gaps between primes

TL;DR

This paper generalizes recent results on bounded prime gaps and large prime gaps, addressing a longstanding question of whether the ratio of consecutive prime gaps can be arbitrarily small or large infinitely often.

Contribution

It proves a unified result extending Maynard-Tao's work and Erdős-Rankin bounds, resolving a 60-year-old problem about the ratios of consecutive prime gaps.

Findings

The ratio of consecutive prime gaps can be arbitrarily small infinitely often.

The ratio of consecutive prime gaps can be arbitrarily large infinitely often.

Provides a strong affirmative answer to Erdős's long-standing question.

Abstract

In the present work we prove a common generalization of Maynard-Tao's recent result about consecutive bounded gaps between primes and on the Erd\H{o}s-Rankin bound about large gaps between consecutive primes. The work answers in a strong form a 60 years old problem of Erd\"os, which asked whether the ratio of two consecutive primegaps can be infinitely often arbitrarily small, and arbitrarily large, respectively.