Dice Question Streamline Icon: https://streamlinehq.com

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.

Information Square Streamline Icon: https://streamlinehq.com

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, 3 Dec 2024) in Section “Orbits of thin groups more generally”