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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.

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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”