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Given a reduced fraction c∕d
with 0 < c < d, there is a unique continued fraction expansion of c∕d as [0;a1,a2,…ar]
with r ≥ 1, 0 ≤ aj for 1 ≤ j ≤ r, and ar ≥ 2. For fixed positive integer n, the
asymptotic distribution of the pair (c,d) mod n among the n2 ∏
p|n(1 − 1∕p2)
possible pairs of congruence classes is uniform when averaged over the set
Q(x) := {(c,d) : 0 < c < d ≤ x,gcd(c,d) = 1} as x →∞. The main result is that if
attention is restricted to the (rather thin) subset Qm(x) of relatively prime pairs
(c,d) so that all the continued fraction convergents aj ≤ m, the same equidistribution
holds. As a corollary, the relative frequency, both in Q(x) and in Qm(x) for any fixed
m > 1, of reduced fractions (c,d) so that d ≡ b mod n, is asymptotic to
n−1 ∏
p| gcd(b,n)(1 − p−1)∏
q|n(1 − q−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.
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