Mathematical Proceedings of the Cambridge Philosophical Society



The h-spacing distribution between Farey points


VOLKER AUGUSTIN a1, FLORIN P. BOCA a2 1 , CRISTIAN COBELI a3 and ALEXANDRU ZAHARESCU a4a5
a1 Richter Weg 12d, 92421 Schwandorf, Germany. e-mail: snowbird@bigfoot.de
a2 School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF2 4YH. e-mail: BocaFP@cardiff.ac.uk On leave from: Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest 70700, Romania
a3 Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest 70700, Romania. e-mail: ccobeli@stoilow.imar.ro
a4 Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest 70700, Romania. e-mail: zaharesc@math.uiuc.edu
a5 Institute for Advanced Study, School of Mathematics, Olden Lane, Princeton, NJ 08540, U.S.A. e-mail: zaharesc@math.uiuc.edu

Abstract

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

and order increasingly its elements γj = aj/qj as α [less-than-or-eq, slant] γ1 < γ2 < … < γNI(Q) [less-than-or-eq, slant] β. The number of elements of [script F]I(Q) is

We simply let [script F](Q) = [script F][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 [rightward arrow] [infty infinity]. 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].

(Received November 16 1999)



Footnotes

1 Work of F. P. B. supported by an EPSRC Advanced Fellowship.