THE DISTRIBUTION MOD n OF FRACTIONS WITH BOUNDED PARTIAL QUOTIENTS

Given a reduced fraction c/d with 0 1, 0 2. For fixed positive integer n, the asymptotic distribution of the pair (c, cf) mod n among the rcΠ. (1 ~ 1/P) possible pairs of congruence classes is uniform when averaged over the set Q(x) := {(c,d) : 0 1, of reduced fractions (c, d) so that d = b mod n, is asymptotic to n ^ p ^ ^ l p ^ Π . i n ί 1 ^ 2 ) 1 . These results lend further heuristic support to Zaremba's conjecture, which in this terminology reads that for some m (perhaps even m = 2) the set of denominators d occurring in Qm(x) includes all but finitely many natural numbers. The proofs proceed from some recent estimates for the asymptotic size of Qm(x). Thereafter, the argument is combinatorial.