Radical bound for Zaremba's conjecture

Abstract

Famous Zaremba's conjecture (1971) states that for each positive integer $q⩾2$, there exists a positive integer $1⩽a<q$, coprime to $q$, such that if you expand a fraction $a/q$ into a continued fraction $a/q=[a_1,\text{…},a_n]$, all of the coefficients $a_i$’s are bounded by some absolute constant $k$, independent of $q$. Zaremba conjectured that this should hold for $k=5$. In 1986, Niederreiter proved Zaremba's conjecture for numbers of the form $q=2^n,3^n$ with $k=3$ and for $q=5^n$ with $k=4$. In this paper, we prove that for each number $q\ne 2^n,3^n$, there exists $a$, coprime to $q$, such that all of the partial quotients in the continued fraction of $a/q$ are bounded by $rad\left(q\right)-1$, where $rad\left(q\right)$ is the radical of an integer number, that is, the product of all distinct prime numbers dividing $q$. In particular, this means that Zaremba's conjecture holds for numbers $q$ of the form $q=2^n{3}^m,n,m\in N\cup \left\{0\right\}$ with $k=5$, generalizing Neiderreiter's result. Our result also improves upon the recent result by Moshchevitin, Murphy and Shkredov on numbers of the form $q=p^n$, where $p$ is an arbitrary prime and $n$ is sufficiently large.