A new bound concerning Zaremba’s conjecture

By the Euclidean algorithm, any rational number $a/q$ can be uniquely represented as a regular continued fraction. In the theory of continued fractions, in 1971 Zaremba famously made the following conjecture: there is an absolute constant $k$ such that for any positive integer $q$, there exists $a$, which is coprime to $q$, such that in the continued fraction expansion of $a/q$, all partial quotients are less than or equal to $k$. Zaremba himself conjectured that the value of this absolute constant $k$ equals 5. Two decades later, Hensley conjectured that for a large, prime denominator, a bound equal to 2 is sufficient. Ever since, further tightening this bound has been an ongoing problem in the number theory community.

In this talk and drinks, Prof. Ilya Shkredov presents a marked improvement to the well-known Korobov bound on this conjecture. In 1963, using exponential sums, Korobov proved that for any prime $q$, there exists a coprime numerator such that the partial quotients never exceed log($q$). Shkredov presents a proof that, for any sufficiently large prime $q$, the partial quotients are bounded by $O$(log $q$/ log log $q$).

Event information

This event is at 5pm on Wednesday, 29 March, on the second floor of the Royal Institution. Grab a drink and say hi before the talk starts at 5:15 in Tyndall’s Parlour. After the talk, everyone meets for discussion over drinks in the Old Post Room. To attend, email at@lims.ac.uk.