Date of Award

2026

Abstract

We establish a hidden stability in the continued fraction expansions of certain real sequences. Classical theory predicts only statistical regularities for the partial quotients of actually random sequences, and repeated applications of the Gauss map would seem to have a randomizing effect even on analytically defined sequences; nevertheless we show how partial quotients connected to sequences governed by asymptotic expansions with rational coefficients always display a certain kind of periodic regularity and, upon taking closures, also an unexpected topological character. Our story connects three types of related data: sequences of real numbers, sequences of continued fraction expansions, and formal Laurent series with rational coefficients. We define a novel formal Gauss map for formal Laurent series with rational coefficients and show how their formal partial quotients turn out quasi-polynomial. Even when the series do not converge, formal Laurent series can govern real sequences as asymptotic expansions but asymptotics will only determine sequences up to an equivalence relation. Nevertheless we show that we can still read off long-term continued fraction structure. Indeed, from these foundations, we establish that the numerical partial quotients coming from the sequences do eventually match the quasi-polynomials that arise at first only formally from the Laurent series. Finally, we explore the algebraic structure of a set of irrational sequences whose partial quotients are eventually quasi-polynomials. By introducing an asymptotic equivalence relation, we construct an injective map that embeds the resulting quotient set into the formal ring: the space of continuous functions from the profinite integers to the field of formal Laurent series over the rationals. This has a very concrete implication: generically, two sequences with structured continued fractions will sum and multiply to sequences with structured continued fractions. This counters the popular intuition that continued fraction arithmetic is by nature opaque. We conclude by proposing conjectures on the uniform degree growth and the unbounded partial quotients of algebraic sequences, and comment on ongoing work and directions for future research.

Document Type

Dissertation

First Advisor

David V Feldman

Second Advisor

Donald Hadwin

Third Advisor

Junhao Shen

Department or Program

Mathematics

Degree Name

Doctor of Philosophy

Download

Share

COinS