N. Moshchevitin

B. Murphy

I. Shkredov

Using methods related to the Bourgain–Gamburd machine refines the previous bound on Zaremba’s conjecture in the theory of continued fractions.

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.

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We prove in particular that for any sufficiently large prime $p$ there is $1\leq a \leq p$ such that all partial quotients of $a/p$ are bounded by $O(\log p/\log \log p)$. For composite denominators a similar result is obtained. This improves the well--known Korobov bound concerning Zaremba's conjecture from the theory of continued fractions.

On Korobov bound concerning Zaremba's conjecture