The averaged Zaremba counting conjecture

For $A\geq 2$, let

$$E_A=\{x=[0;a_1,a_2,\ldots]\in[0,1]:a_j\leq A\},$$

and define

$$\Sigma_{N,A}=\left\{\frac{a}{N}\in E_A:1\leq a<N,\ (a,N)=1\right\},$$ $$\Omega_{N,A}=\left\{\frac{a}{n}\in E_A:1\leq a<n\leq N,\ (a,n)=1\right\}.$$

The averaged Zaremba conjecture. For $N\in\mathbb{N}$,

$$|\Sigma_{N,A}|\sim\frac{N^{2\delta_A}}{N}=N^{2\delta_A-1}.$$

Here $\delta_A$ is the Hausdorff dimension of $E_A$, and $f\sim g$ means $f/g\to1$ as the relevant variable tends to infinity. This interpretation is motivated by Hensley's observation that $|\Omega_{N,A}|\sim N^{2\delta_A}$; the supplied text does not specify whether the asserted asymptotic has been proved.