Zaremba’s conjecture belongs to the area of continued fractions. It predicts that for any given positive integer q there is a positive a, a less than q, (a,q)=1 such that all partial quotients bj in its continued fractions expansion a/q = 1/b1+1/b2 +… + 1/bs are bounded by five. At the moment the question is widely open although the area has a rich history of works by Korobov, Hensley, Niederreiter, Bourgain and many others. We survey certain results concerning this hypothesis and show how growth in groups helps to solve different relaxations of Zaremba’s conjecture. In particular, we show that a deeper hypothesis of Hensley concerning some Cantor-type set with the Hausdorff dimension strictly greater than 1/2 takes place for the so-called modular form of Zaremba’s conjecture.
This video is part of the Number Theory Web Seminar series.
