Zaremba’s Conjecture (bounded partial quotients for all denominators)

Determine whether there exists a universal constant A > 0 such that, for every integer u ≥ 1, there is an integer t with 1 ≤ t < u and gcd(t, u) = 1 whose regular continued fraction expansion t/u = [a1, …, aℓ] has all partial quotients bounded by A.

Background

Zaremba’s conjecture is a classical problem in Diophantine approximation. Bourgain–Kontorovich showed that the conjecture holds for a positive proportion of denominators, and subsequent work has improved the threshold on the Hausdorff dimension required for these positive proportion results. The full ‘for all u’ statement remains open.