Continued Fractions: An Arithmetic and Analytic Study




Handley, Hunter

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Contemporary research in continued fractions has motivated novel results in spectral analysis— specifically in the characterization of doubly palindromic discrete m- functions. In this thesis, we first discuss historical results in the field of continued fractions by the likes of Euler, Lagrange, and Galois. Next, we discuss an illuminating theorem by Serret before introducing the work of Dr. Edward Burger who classified lattice equivalences via continued fractions. This result is the exact motivation for the analog we provide in spectral m-functions later. The following chapter is a thorough discussion of spectral theory & analysis, which will lay out the background necessary to understand the novel results presented. Topics such as Herglotz functions, discrete m-functions, and J-fractions will be included, along with prior results by Derevyagin et al. and Barry Simon. In the final chapter, we define the tools necessary to state completely novel results: a discrete m-function is congruent to its functional conjugate if and only if its J-fraction expansion has a tail of doubly palindromic Jacobi parameters. This represents the preprint by Handley and Simanek.



Spectral theory.