Limit points of the sequence of normalized differences between consecutive prime numbers

Abstract

Let pn denote the nth prime number and let dn = pn+1-pn denote the nth difference in the sequence of prime numbers. Erdos and Ricci independently proved that the set of limit points of dn/log pn, the normalized differences between consecutive prime numbers, forms a set of positive Lebesgue measure. Hildebrand and Maier answered a question of Erdos and proved that the Lebesgue measure of the set of limit points of dn/log pn in the interval [0, T] is » T as T → 1. Currently, the only specific limit points known are 0 and1. In this note, we use the method of Erdos to obtain specific intervals within which a positive Lebesgue measure of limit points exist. For example, the intervals [1/8,2] and [1/40,1] both have a positive Lebesgue measure of limit points.