On Korobov bound concerning Zaremba's conjecture

We prove in particular that for any sufficiently large prime $p$ there is $1\le a<p$ such that all partial quotients of $a/p$ are bounded by $O(\log p/\log \log p)$. This improves the well–known Korobov bound (1963) concerning Zaremba's conjecture from the theory of continued fractions. In our talk we will discuss the scheme of the proof, the connection with non-commutative methods in number theory and with growth in groups, as well as further possible directions of research.
Conference ID: 942 0186 5629 Password is a six-digit number, the first three digits of which form the number p + 44, and the last three digits are the number q + 63, where p, q is the largest pair of twin primes less than 1000