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].

The distribution on the torus ℝ/ℤ of a set of fractions of the form

(formula here)

is investigated, where q is a large integer, m is the inverse of m modulo q, R(x) is a rational function defined modulo q, and [Uscr], [Mscr], [Nscr] are subsets of {1,…,q}. Under some natural assumptions, it is shown that the set [Rscr] is uniformly distributed on ℝ/ℤ.

Let q be a prime number and let a = (a1, …, as) be an s-tuple of distinct integers modulo q. For any x coprime with q, let be such that . For fixed s and q→∞ an asymptotic formula is given for the number of residue classes x modulo q for which

The more general case, when q is not necessarily prime and x is restricted to lie in a given subinterval of [1, q], is also treated.