Zaremba’s conjecture on bounded partial quotients

Establish the existence of an absolute constant Z such that every positive integer occurs as the denominator of a reduced rational number whose continued-fraction partial quotients are all at most Z.

Background

The conjecture connects Diophantine properties of rationals with thin-orbit dynamics of semigroups generated by continued-fraction matrices. Significant progress shows positive-proportion results for many bounds Z, but the uniform all-denominators statement remains open.

The notes discuss the thin-orbit framework and related obstructions in bounded partial quotient sets.

References

Another famous conjecture is part of the same general type. There exists a positive constant Z such that every natural number is the denominator of some rational number (in reduced form) whose continued fraction partial quotients are ≤ Z.

— An illustrated introduction to the arithmetic of Apollonian circle packings, continued fractions, and other thin orbits  (2412.02050 - Stange, 2024) in Section “Orbits of thin groups more generally”

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]