Limit points and long gaps between primes (Q2814321)

Let \(_n= p_{n+1}- p_n\) with \(p_n\) the \(n\)th prime. Various authors have studied the sequence \(({d_n\over\log p_n})\) and its set of limit points. Whilst \(0\), \(\infty\) are the only known limit points of this sequence, \textit{W. D. Banks}, the second author and \textit{J. Maynard} proved in [``On limit points of the sequence of normalized prime gaps'', Preprint, \url{arxiv:1404.5094}] that at least 12.5\% of nonnegative real numbers are limit points. Recently other authors have investigated the limit points of the sequence \(({d_n\over R(p_n)})\), where \(R(T)= \log T\log_2T\log_4 T(\log_3 T)^{-2}\) with \(\log_2T= \log\log T\) and so on. Building on the work in the paper cited above combined with that of \textit{K. Ford} et al. [Ann. Math. (2) 183, No. 3, 935--974 (2016; Zbl 1338.11083)], \textit{J. Pintz} in [``On the distribution of gaps between consecutive primes'', Preprint, \url{arxiv:1407.2213}] and in [``A note on the distribution of normalized prime gaps'', Preprint, \url{arxiv:1510.04577}] established that at least 25\% of nonnegative real numbers are limit points of the sequence \(({d_n\over R(p_n)})\) and he showed that a similar result holds when \(R(T)\) is replaced by certain other functions. The aim of the current paper is to integrate the ideas of all the authors cited above and to obtain more general results with \(R(T)\) replaced by any `reasonable' function as defined in the paper. A special case of their main result replaces \(R(T)\) by \(R_1(T)= \log T\log_2T/\log_3T\). Let \(L_{R_1}\) denote the set of limit points of \(({d_n\over R_1(p_n)})_{p_n\geq T_0}\), where \(\log_3 T_0\geq 1\).NEWLINENEWLINE Then Theorem 1.1 states that for any nonnegative real numbers \(\alpha_1,\dots,\alpha_5\) with \(\alpha_1\leq\alpha_2\leq\cdots\leq \alpha_5\) we have \(\{\alpha_j- \alpha_i: 1\leq i< j\leq 5\}\cap L_{R_1}\neq\emptyset\). The authors deduce that at least 25\% of nonnegative real numbers belong to \(L_{R_1}\).NEWLINENEWLINE Theorem 6.2 (not stated here) concerns the set of limit points \(L_f\) of the sequence \(({d_n\over f(p_n)})\) for \(f\) a `reasonable' function. From this theorem a conditional result stating that \(L_{R_1}\) may contain at least 33\({1\over 3}\)\% or even 50\% of nonnegative real numbers is deduced. The authors also apply the techniques to chains of consecutive gaps between primes.