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    Heersink, Byron 2016. Poincaré sections for the horocycle flow in covers of $$\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})$$ SL ( 2 , R ) / SL ( 2 , Z ) and applications to Farey fraction statistics. Monatshefte für Mathematik, Vol. 179, Issue. 3, p. 389.

    Kunik, Matthias 2016. A scaling property of Farey fractions. European Journal of Mathematics, Vol. 2, Issue. 2, p. 383.

    Polanco, Geremías Schultz, Daniel and Zaharescu, Alexandru 2015. Continuous distributions arising from the Three Gap Theorem. International Journal of Number Theory, p. 1650107.

    Alkan, Emre Xiong, Maosheng and Zaharescu, Alexandru 2008. Quotients of values of the Dedekind Eta function. Mathematische Annalen, Vol. 342, Issue. 1, p. 157.

    Alkan, Emre Xiong, Maosheng and Zaharescu, Alexandra 2007. Local spacings along curves. Journal of Mathematical Analysis and Applications, Vol. 329, Issue. 1, p. 721.

    Boca, Florin P. and Zaharescu, Alexandru 2006. The Distribution of the Free Path Lengths in the Periodic Two-Dimensional Lorentz Gas in the Small-Scatterer Limit. Communications in Mathematical Physics, Vol. 269, Issue. 2, p. 425.

    Haynes, Alan 2003. A note on Farey fractions with odd denominators. Journal of Number Theory, Vol. 98, Issue. 1, p. 89.

  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 131, Issue 1
  • July 2001, pp. 23-38

The h-spacing distribution between Farey points

  • DOI:
  • Published online: 26 October 2001

Let I = [α, β] be a subinterval of [0, 1]. For each positive integer Q, we denote by [Fscr ]I(Q) the set of Farey fractions of order Q from I, that is

and order increasingly its elements γj = aj/qj as α [les ] γ1 < γ2 < … < γNI(Q) [les ] β. The number of elements of [Fscr ]I(Q) is

We simply let [Fscr ](Q) = [Fscr ][0,1](Q), N(Q) = N[0,1](Q).

Farey sequences have been studied for a long time, mainly because of their role in problems related to diophantine approximation. There is also a connection with the Riemann zeta function which has motivated their study. Farey sequences seem to be distributed as uniformly as possible along [0, 1]; a way to prove it is to show that

for all ε > 0, as Q → ∞. Yet this is a very strong statement, as Franel and Landau [3, 4] have shown that (1·3) is equivalent to the Riemann Hypothesis.

Our object here is to investigate the distribution of spacings between Farey points in subintervals of [0, 1]. Various results related to this problem have been obtained by [2, 3, 5–8, 10–13].

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
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