Gaps Between Prime Numbers

Abstract

Let ${d_n} = {p_{n + 1}} - {p_n}$ denote the $n$th gap in the sequence of primes. We show that for every fixed integer $k$ and sufficiently large $T$ the set of limit points of the sequence $\{ ({d_n}/\log n, \ldots ,{d_{n + k - 1}}/\log n)\}$ in the cube ${[0,T]^k}$ has Lebesgue measure $\geq c(k){T^k}$, where $c(k)$ is a positive constant depending only on $k$. This generalizes a result of Ricci and answers a question of Erdös, who had asked to prove that the sequence $\{ {d_n}/\log n\}$ has a finite limit point greater than 1.