Large gaps between consecutive prime numbers

@article{Ford2014LargeGB,
  title={Large gaps between consecutive prime numbers},
  author={Kevin Ford and Ben Green and Sergei Konyagin and Terence Tao},
  journal={arXiv: Number Theory},
  year={2014}
}
Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdos, we show that $$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},$$ where $f(X)$ is a function tending to infinity with $X$. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions… 
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We show that there exists pairs of consecutive primes less than $x$ whose difference is larger than $t(1+o(1))(\log{x})(\log\log{x})(\log\log\log\log{x})(\log\log\log{x})^{-2}$ for any fixed $t$. Our…
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The main new ingredient is a generalization of a hypergraph covering theorem of Pippenger and Spencer, proven using the R\"odl nibble method.
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