Radical bound for Zaremba's conjecture (Q6593636)

The author makes a thorough study of the developments concerning \textit{S. K. Zaremba}'s conjecture [in: Appl. Number Theory Numer. Analysis, Proc. Sympos. Univ. Montreal 1971, 39--119 (1972; Zbl 0246.65009)]:\N\Nfor each positive integer \(q\geq 2\), there exists a positive integer \(1\leq a < q\), coprime to \(q\), such that if you expand \(a/q\) into a continued fraction \(a/q=[a_1,\ldots,a_n]\), all of the coefficients of the \(a_i\)'s are bounded by some absolute constant \(\mathcal{T}\), independent of \(q\).\N\NThe layout of the paper is as follows:\N\N\S1. Introduction and main result (\(2\) pages).\N\NThis contains a short history of the developments with respect to (partial) proofs of the conjecture and gives the main theorem of the paper:\N\NTheorem 1.2. For any integer \(q\geq 2\), such that \(q\not= 2^n,3^n\), there exists a positive integer \(a\) with \(1\leq a< q\) and \(\text{gcd}(a,q)=1\), such that \(K(a/q)\leq\text{rad}(q)-1\) (here \(\text{rad}(n)=\prod_{p|n,p\,\text{prime}}\ p\)).\N\N\S2. Proof of the main result (\(5\) pages).\N\NReferences (\(19\) items).