Error approximation for backwards and simple continued fractions

Abstract

In this paper, we provide a new framework for studying continued fractions (CFs) by means of the backwards continued fraction (BCF). We develop an approximation theory for BCFs based on taking expansions of a fixed length, show the correspondence between continued fractions and their BCFs counterpart, and illustrate a rich approximation theory for continued fractions based off the methods of the approximation theory for the backwards case. In particular, we construct explicit functions that are sharp bounds for the BCF or CF error infinitely often over any BCF or CF cylinder set, and work out the details to pass seamlessly between the BCF and CF expansion of any real number.