Chains of Large Gaps Between Primes

Let p n denote the n-th prime, and for any k ≥ 1 $$k \geqslant 1$$ and sufficiently large X, define the quantity G k ( X ) : = max p n + k ≤ X min ( p n + 1 − p n , … , p n + k − p n + k − 1 ) , $$\displaystyle G_k(X) := \max _{p_{n+k} \leqslant X} \min ( p_{n+1}-p_n, \dots , p_{n+k}-p_{n+k-1} ), $$ which measures the occurrence of chains of k consecutive large gaps of primes. Recently, with Green and Konyagin, the authors showed that G 1 ( X ) ≫ log X log log X log log log log X log log log X $$\displaystyle G_1(X) \gg \frac {\log X \log \log X\log \log \log \log X}{\log \log \log X} $$ for sufficiently large X. In this note, we combine the arguments in that paper with the Maier matrix method to show that G k ( X ) ≫ 1 k 2 log X log log X log log log log X log log log X $$\displaystyle G_k(X) \gg \frac {1}{k^2} \frac {\log X \log \log X\log \log \log \log X}{\log \log \log X} $$ for any fixed k and sufficiently large X. The implied constant is effective and independent of k.