On limit points of the sequence of normalized prime gaps

Let pnpn denote the nnth smallest prime number, and let LL denote the set of limit points of the sequence {(pn+1−pn)/logpn}∞n=1{(pn+1−pn)/log⁡pn}n=1∞ of normalized differences between consecutive primes. We show that, for any sequence of nine nonnegative real numbers β1≤β2≤⋯≤β9β1≤β2≤⋯≤β9, at least one of the numbers βj−βiβj−βi (1≤i