On the intervals of a third between Farey fractions

The spacing distribution between Farey points has drawn attention in recent years. It was found that the gaps γj+1 -γj between consecutive elements of the Farey sequence produce, as Q →∞, a limiting measure. Numerical computations suggest that for any d ≥ 2, the gaps ~γj+d -γj 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.