On the index of Farey sequences

Abstract. We construct a Poincaré section for the horocycle flow on the modular surface SL(2, R)/SL(2, Z), and study the associated first return map, which coincides with a transformation (the BCZ map) defined by Boca-Cobeli-Zaharescu [8]. We classify ergodic invariant measures for this map and prove equidistribution of periodic orbits. As corollaries, we obtain results on the average depth of cusp excursions and on the distribution of gaps for Farey sequences and slopes of lattice vectors.

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“…Applying Theorem 1.3 to κ α (note that α ∈ (0, 2) implies that κ α ∈ L 1 ), we obtain a result originally due to Boca-Gologan-Zaharescu [10]:…”

Section: 53mentioning

confidence: 68%

A Poincaré Section for the Horocycle Flow on the Space of Lattices

Athreya

Cheung

2013

International Mathematics Research Notices

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“…The second step in the proof of (1.1) in [2] relies on [2, Lemma 1], which is actually exactly Remark 2.6 in [6] (see also [4,Lemma 5]), and on a result relating the ℓindex of a Farey fraction and the continuant of regular continued fractions. The ℓ-index of γ i = ai qi ∈ F Q is the positive integer ν ℓ (γ i ) = a i+ℓ−1 q i−1 − a i−1 q i+ℓ−1 where a i+k q i+k denotes the k th successor of γ i in F Q .…”

Section: The Index and The Continuantmentioning

confidence: 99%

Gap Distribution of Farey Fractions Under Some Divisibility Constraints

Boca¹,

Heersink²,

Spiegelhalter³

2014

Integers

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“…Note that here we are using Lemma 3 and the fact that QT k ∩ Z 2 vis = ∅ for k > 2Q. Now we estimate each of these sums using the following classical result which we quote from [4,Corollary 2.2].…”

Section: Proof Of Theoremmentioning

confidence: 99%

Numerators of Differences of Nonconsecutive Farey Fractions

Haynes

2010

Int. J. Number Theory

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Abstract. An elementary but useful fact is that the numerator of the difference of two consecutive Farey fractions is equal to one. For triples of consecutive fractions the numerators of the differences are well understood and have applications to several interesting problems. In this paper we investigate numerators of differences of fractions which are farther apart. We establish algebraic identities between such differences which then allow us to calculate their average values by using properties of a measure preserving transformation of the Farey triangle.

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On the index of Farey sequences

Abstract: We prove some asymptotic formulae concerning the distribution of the index of Farey fractions of order Q as Q → ∞.

Cited by 14 publications

References 5 publications

A Poincaré Section for the Horocycle Flow on the Space of Lattices

A Poincaré Section for the Horocycle Flow on the Space of Lattices

Gap Distribution of Farey Fractions Under Some Divisibility Constraints

Numerators of Differences of Nonconsecutive Farey Fractions

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