Finiteness of exceptions for small bounds in Zaremba-type problems

Determine whether for Z=3 (respectively Z=2) only finitely many positive integers fail to occur as denominators of reduced rationals whose continued-fraction partial quotients are bounded by Z.

Background

The conjectural finiteness of exceptions for small bounds would sharply strengthen Zaremba’s conjecture in specific cases. While major advances exist for positive density and almost-all results, complete finiteness for Z=3 or Z=2 remains unsettled.

These finiteness statements were specifically proposed by Niederreiter (Z=3) and Hensley (Z=2).

References

Niederreiter conjectured that even for Z=3 there are only finitely many exceptions; Hensley conjectured this for Z=2.

— 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”