Analytic and ergodic properties of certain classes of Euclidean algorithms

Siskaki, Maria

Permalink

https://hdl.handle.net/2142/120256 Copy

Description

Title

Analytic and ergodic properties of certain classes of Euclidean algorithms

Author(s)

Siskaki, Maria

Issue Date

2023-04-21

Director of Research (if dissertation) or Advisor (if thesis)

Boca, Florin

Doctoral Committee Chair(s)

Zaharescu, Alexandru

Committee Member(s)

• Rosenblatt, Joseph
• Berndt, Bruce
• Athreya, Jayadev

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Euclidean algorithms
• continued fractions
• quadratic irrationals
• Kloosterman sums
• Gauss conjecture
• Perron-Frobenius operator
• Farey sequence
• pair correlation

Abstract

This thesis is concerned with results on analytic and ergodic properties of certain Continued Fraction Gauss maps arising from various Euclidean division algorithms. First, we study the distribution of the periodic points of the Even, Backwards, and Odd Gauss type shifts. These points coincide with certain classes of reduced quadratic irrationals, and, for the Regular Continued Fraction Gauss map, are known \cite{Fa,Po} to be uniformly distributed with respect to the Gauss measure. Here, we adapt a number theoretical approach initiated in \cite{KO} and fully developed in \cite{Bo} to show that the $E$-, $B$-, and $O$-periodic points are uniformly distributed with the respect to the corresponding invariant measures. This approach ultimately relies on the application of Weil bounds on Kloosterman sums, and therefore provides an effective error in the final asymptotic formulas. It is worth noting that the invariant measures of the Even and Backwards Continued Fractions Gauss maps are infinite, making a Perron-Frobenius functional analytic approach as in \cite{Fa,Po} more complex. Second, we study the Gauss-Kuzmin-L\'evy problem for the Nearest Integer Continued Fraction Gauss map through a functional analytic approach. In particular, we provide effective formulas for the fact that the probability of landing in $[0,x]$ after successive applications of the NICF Gauss map $T$ is asymptotically equal to the $T$-invariant measure of $[0,x]$. The constant giving the speed of convergence in this effective formula is smaller than the (optimal) Wirsing constant for the Regular Continued Fraction Gauss map case. Finally, we study the local statistics for sequences of rationals involved in the Diophantine approximation of regular reduced quadratic irrationals. In particular, the pair correlation functions of Farey fractions with denominators $q$ satisfying $(q, m) = 1$, respectively $q \equiv b \mod{m}$ with $(b, m) = 1$, are shown to exist and are explicitly computed.

Graduation Semester

2023-05

Type of Resource

Thesis

Copyright and License Information

Copyright 2023 Maria Siskaki

Owning Collections