Hensley’s conjecture: dimension threshold versus coverage of denominators

Establish that the Hausdorff dimension δ_A of the set E_A = { x ∈ [0,1] : all continued fraction digits of x are ≤ A } satisfies δ_A > 1/2 if and only if D_A contains all sufficiently large integers, where D_A = { N ∈ ℕ : there exists a ∈ (ℤ/Nℤ)× with a/N ∈ E_A }.

Background

Hensley proposed a precise connection between the Hausdorff dimension δ_A of the bounded-digit continued fraction Cantor set E_A and the validity of Zaremba’s conjecture for all sufficiently large denominators.

Under this conjecture, the critical threshold δ_A > 1/2 is equivalent to the set of admissible denominators D_A eventually containing all positive integers, i.e., Zaremba’s bounded-digit property holds for all sufficiently large N. This ties fractal dimension to a number-theoretic completeness phenomenon.

References

Further, Hensley claimed a precise connection between the Hausdorff dimension of $E_A$ and Zaremba's conjecture as follows. We have $\delta_A>1/2$ if and only if $D_A \supset _{\gg 1}$, i.e. Conjecture~\ref{conj:Z} takes place for all sufficiently large $N$.

— Asymptotic statistics for finite continued fractions with restricted digits (2512.11357 - Lee, 12 Dec 2025) in Section 1 (Introduction), Conjecture [Hensley]