Maseeh Mathematics + Statistics Colloquium Series presents: Regular, Semi-Regular, and Proper Continued Fractions

Title: Regular, Semi-Regular, and Proper Continued Fractions

Abstract: There are numerous equivalent derivations of the regular continued fraction of an irrational number. We begin with a review of the interpretation of the Gauss map as the (normalized length) rotation given by the first-return map of rotation by x (modulo one) on an interval of length 1-x. In joint work with N. Langeveld we develop similar schemes to produce both the semi-regular (numerators plus or minus one) and proper continued fraction (numerators arbitrary positive integers) representations of x. In both of these variants an irrational x has uncountably many such representations; our construction proposes a natural parameterization of each. We also give a characterization of which positive integers may appear as a denominator for proper continued fractions as an analogue to the classical characterization of regular continued fraction convergents as best approximations of the second kind.

Biography: David Ralston is an Associate Professor in the Mathematics department at SUNY Old Westbury, where he has taught since 2012. Before that he held postdoctoral positions at Ben Gurion University of the Negev and The Ohio State University, and he received his Ph.D. in 2008 at Rice University under Professor Michael Boshernitzan. His research focuses on applying techniques in ergodic theory to topics in number theory, especially renormalization techniques to the study of continued fractions. Other topics of research include geodesic flows on infinite translation surfaces, topics in symbolic dynamics, and studying growth rates of ergodic sums in certain skew product systems. He has also improved the materials available to his undergraduate students, producing lecture notes, recordings, and other resources for several courses.