Submit progress

on Zaremba’s conjecture constant ($C_{58}$)

The statement

Zaremba’s conjecture concerns denominators of rational numbers $b/d\in(0,1)$ whose finite continued fraction expansions have all partial quotients bounded by an absolute constant. [Kan2021-conj-reformulation]

Equivalently (in the standard notation of the subject), the conjecture predicts that there exists an integer $A>1$ such that every positive integer appears among the denominators that admit a continued-fraction expansion with all partial quotients in $\{1,2,\ldots,A\}$ . [BK2014-conj-1.1-A5]

We define

C_{58}\ :=\ A_{\mathrm{Zar}},

where $A_{\mathrm{Zar}}$ is the least integer $A$ (if it exists) such that every positive integer $d$ occurs as a denominator of some reduced fraction $b/d$ whose finite continued fraction has all partial quotients $\le A$ . If no such $A$ exists, set $A_{\mathrm{Zar}}:=+\infty$ .

Zaremba conjectured that $A=5$ suffices, and moreover the $A=4$ version is known to be false (explicit counterexamples $d=54$ and $d=150$ are known), giving the rigorous lower bound

A_{\mathrm{Zar}}\ \ge\ 5.

[Kan2021-A4-counterexamples]

The strongest proven result toward finiteness is that the conjecture holds for a density-one set of denominators: Bourgain–Kontorovich prove a density-one statement and record an explicit choice $A=50$ for it. [BK2014-density-one] [BK2014-A50-suffices]

Huang later improved the density-one statement to partial quotients bounded by $5$ (still for a density-one subset of denominators, not all integers). [Hua2015-density-one-A5]

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.