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].