Bounding Zaremba’s conjecture

On Korobov bound concerning Zaremba's conjecture

Zaremba’s conjecture states that every integer appears as the denominator of a finite continued fraction whose partial quotients are bounded by an absolute constant, and that this constant is 5. Tightening this bound has been an open problem in number theory for decades. Using methods related to the Bourgain–Gamburd expansion machine, we obtain a marked improvement to the well-known Korobov bound on this constant.