On the Intervals of a Third between Farey Fractions

Details On the Intervals of a Third between Farey Fractions On the Intervals of a Third between Farey Fractions

2005-11-14

arxiv.org/abs/math/0511363v1

Mathematics

Number Theory

12 pages, one figure

Scientific paper

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.

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