On limit points of the sequence of normalized prime gaps

Abstract: Let $p_n$ denote the $n$th smallest prime number, and let $\boldsymbol{L}$ denote the set of limit points of the sequence ${(p_{n+1} - p_n)/\log p_n}_{n = 1}^{\infty}$ of normalized differences between consecutive primes. We show that for $k = 9$ and for any sequence of $k$ nonnegative real numbers $\beta_1 \le \beta_2 \le ... \le \beta_k$, at least one of the numbers $\beta_j - \beta_i$ ($1 \le i < j \le k$) belongs to $\boldsymbol{L}$. It follows at least $12.5%$ of all nonnegative real numbers belong to $\boldsymbol{L}$.