Averaged Zaremba conjecture for the count |Σ_{N,A}|

Determine the asymptotic formula |Σ_{N,A}| ∼ N^{2δ_A − 1}, where Σ_{N,A} = { a/N ∈ E_A : 1 ≤ a < N, gcd(a, N) = 1 } and δ_A = dim_H(E_A) is the Hausdorff dimension of the set of numbers in [0,1] whose continued fraction digits are at most A.

Background

Building on Hensley’s observation that the aggregated set Ω_{N,A} of rationals with bounded digits up to denominator N has size ~ N^{2δ_A}, the paper formulates an averaged interpretation of Zaremba’s conjecture for fixed denominators.

The conjectured asymptotic |Σ_{N,A}| ∼ N^{2δ_A−1} predicts the typical number of reduced fractions a/N with bounded partial quotients and directly incorporates the fractal dimension δ_A into the counting exponent.