Generalized continued fraction expansions with constant partial denominators

Abstract

<div lang="en" class="abs"><p class="abs_header">Abstract</p><p>We study generalized continued fraction expansions of the form \[\begin{eqnarray}\frac{a_{1}}{N}\frac{}{+}\frac{a_{2}}{N}\frac{}{+}\frac{a_{3}}{N}\frac{}{+}\frac{}{\cdots },\end{eqnarray}\] where \(N\) is a fixed positive integer and the partial numerators \(a_{i}\) are positive integers for all \(i\). We call these expansions \(\operatorname{dn}_{N}\) expansions and show that every positive real number has infinitely many \(\operatorname{dn}_{N}\) expansions for each \(N\). In particular, we study the \(\operatorname{dn}_{N}\) expansions of rational numbers and quadratic irrationals. Finally, we show that every positive real number has, for each \(N\), a \(\operatorname{dn}_{N}\) expansion with bounded partial numerators.</p></div>