On the Intervals of a Third between Farey Fractions

Abstract

The spacing distribution between Farey points has drawn attention in recent years. It was found that the gaps ${\gamma }_{j+1}-{\gamma }_j$ between consecutive elements of the Farey sequence produce, as $Q\to \infty$, a limiting measure. Numerical computations suggest that for any $d\ge 2$, the gaps ${\gamma }_{j+d}-{\gamma }_j$ also produce a limiting measure whose support is distinguished by remarkable topological features. Here we prove the existence of the spacing distribution for $d=2$ and characterize completely the corresponding support of the measure.

12 pages, one figure