Limit Points of the Sequence of Normalized Differences Between Consecutive Prime Numbers

Let p n denote the nth prime number and let d n = p n + 1 − p n $$d_{n} = p_{n+1} - p_{n}$$ denote the nth difference in the sequence of prime numbers. Erdős and Ricci independently proved that the set of limit points of d n log p n $$\frac{d_{n}} {\log p_{n}}$$ , the normalized differences between consecutive prime numbers, forms a set of positive Lebesgue measure. Hildebrand and Maier answered a question of Erdős and proved that the Lebesgue measure of the set of limit points of d n log p n $$\frac{d_{n}} {\log p_{n}}$$ in the interval [0, T] is ≫ T as T → ∞ $$T \rightarrow \infty$$ . Currently, the only specific limit points known are 0 and ∞ $$\infty$$ . In this note, we use the method of Erdős to obtain specific intervals within which a positive Lebesgue measure of limit points exist. For example, the intervals 1 8 , 2 $$\left [\frac{1} {8},2\right ]$$ and 1 40 , 1 $$\left [ \frac{1} {40},1\right ]$$ both have a positive Lebesgue measure of limit points.