The distribution \(\bmod n\) of fractions with unbounded partial quotients
type: Item
instance of: scholarly article
title: The distribution \(\bmod n\) of fractions with unbounded partial quotients
zbMATH Open document ID: 0816.11002
author: Doug Hensley
published in: Pacific Journal of Mathematics
publication date: 1995-06-27T00:00:00Z
review text: This paper concerns (reduced) fractions with bounded partial quotients, and the distribution \(\text{mod } n\) of the corresponding integer pairs (numerator, denominator)=\((a,b)\) say. With or without a bound on the partial quotients, there are certain pairs of congruence classes \(\text{mod } n\) that are forbidden to such pairs, e.g. \((0,0)\). There are \(n^ 2 \prod_{p\mid n} (1-p^{-2})\) possible values for \((a,b)\bmod n\). If there is no constraint on the partial quotients, an elementary inclusion and exclusion argument shows that the asymptotic distribution, taken over all fractions with \(1\leq a<b\leq x\), of \((a,b)\bmod n\) is uniform across all available congruence class pairs. The main result is that, for any \(k>1\), and when attention is restricted to fractions for which all partial quotients are \(\leq k\), this asymptotically uniform distribution persists. The proof requires recently established asymptotic estimates for the number of such fractions with \(1\leq a<b\leq x\) (as a function also of \(k\)), and then a `mixing' argument that takes the place of inclusion and exclusion. The author seems to think (as does the reviewer, who is the author) that this result lends support, with \(k=2\), to Zaremba's conjecture to the effect that there exists \(k\) such that for all sufficiently large \(b\), there exists an \(a\) so that all partial quotients in the continued fraction expansion of \(a/b\) are \(\leq k\).
Mathematics Subject Classification ID: 11A05
Mathematics Subject Classification ID: 11A07
zbMATH DE Number: 759233
zbMATH Keywords: Euclidean algorithm
zbMATH Keywords: fractions with bounded partial quotients
zbMATH Keywords: asymptotic distribution
zbMATH Keywords: Zaremba's conjecture
zbMATH Keywords: continued fraction expansion
MaRDI profile type: Publication
full work available at URL: pjm.1994.166.43
OpenAlex ID: W2002880631
DOI: 10.2140/PJM.1994.166.43
description: scientific article; zbMATH DE number 759233