Zaremba’s conjecture on bounded partial quotients

Establish that for every integer N ≥ 1, there exists an integer a with gcd(a, N) = 1 such that the finite simple continued fraction expansion of the rational number a/N has all partial quotients bounded above by a fixed absolute constant A ≥ 2 that does not depend on N.

Background

Continued fraction expansions of rationals terminate and have integer partial quotients (digits). Zaremba’s conjecture posits the existence, for every denominator N, of a reduced fraction a/N whose finite continued fraction uses only digits up to a universal bound A, independent of N.

Substantial progress has been made: Bourgain and Kontorovich showed the conjecture for A = 50 for almost every N (in a density sense), but the full conjecture remains unresolved. The paper frames this conjecture via the fractal set E_A of real numbers with digits bounded by A and relates it to Hausdorff dimension and transfer operators.

References

One of the famous conjectures in Diophantine approximation, predicted by Zaremba in 1971, concerns analytic structures of finite continued fraction expansions with restricted digits and simply stated as follows. Let $N \in $. Then there exists $a \in (/N)\times$ such that \frac{a}{N}=[0; a_1, a_2, \ldots, a_\ell] has all $a_j \leq A$ for some absolute $A \geq 2$.

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