A new bound concerning Zaremba’s conjecture
5 pm, 29 Mar 2023
Prof. Ilya Shkredov presents a marked improvement to the Korobov bound concerning Zaremba’s conjecture in the theory of continued fractions.
Any irreducible fraction can be uniquely represented as a regular continued fraction. From the theory of continued fractions, Zaremba’s famous 1971 conjecture posits that every integer appears as the denominator of a finite continued fraction whose partial quotients are bounded by an absolute constant. Zaremba himself conjectured that the value of this absolute constant 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 seminar, Prof. Ilya Shkredov presents a marked improvement to the well-known Korobov bound on Zaremba’s conjecture. In 1963, using exponential sums, Korobov proved that for any prime denominator, there exists a coprime numerator such that the partial quotients of the continued fraction expansion never exceed the log of the denominator. Shkredov presents a proof that, for any sufficiently large prime denominator q, all partial quotients never exceed O(log q/log log q). The same result takes place for composite q.
This seminar takes place on Wednesday 29 March 2023 at 5 pm in Tyndall’s Parlour at the London Institute, on the second floor of the Royal Institution in Mayfair. The seminar will be followed by drinks and refreshments onsite. To register to attend, please email email@example.com.
Prof. Ilya Shkredov is an Arnold Fellow at the London Institute. His areas of expertise include generalisations of Szemerédi's theorem, the sum-product phenomenon, inverse results in additive combinatorics, uniformly distributed sequences, and ordinary and multiple recurrence.