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Title:  Beatty ratios, algorithms related to Sturmian sequences and uniform distribution theory 
Author(s):  Polanco Encarnacion, Geremias 
Director of Research:  Stolarsky, Kenneth B. 
Doctoral Committee Chair(s):  Berndt, Bruce C. 
Doctoral Committee Member(s):  Stolarsky, Kenneth B.; Hildebrand, A.J.; Zaharescu, Alexandru 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Beatty Sequence
Sturmian Sequence characteristic Sequence Frullani's integral Steinhaus Theorem Three Gap Theorem Kloosterman Sums Farey Fractions 
Abstract:  This dissertation is divided into three main sections. The main result of Section 1 is that, for
$a,b>1$, irrational, the quantity $\log (a/b)$ is ``not too far'' from the series of fractional parts
$$ \sum_{n=1}^{\infty}\frac{a \{a^{1} (n+1)\}b\{b^{1}(n+1)\}}{n(n+1)} \text{,}$$
i.e, the absolute value of the difference of these two quantities is less than one plus a ``small'' constant $k(a,b)$.
In order to prove this result, we first study
properties of ratios of elements of Beatty sequences and their
connection with Sturmian sequences. If we write $a_n=\lfloor n a
\rfloor$ for $a$ a real number and $n \in \N$, then we are
concerned with ratios of the form
$$r_{a}(n,k):=\frac{ a_{n+k}}{a_n} $$
\noindent as well as their reciprocals. These types of quotients have various
properties. For example, the $i^{th}$ element of a Sturmian sequence with slope $s:=s(a,b)$
can be defined by a difference of ratios $R(i):=r_{a}(i,1)r_{b}(i,1)$
for some irrational numbers $a$, $b$, i.e, $R(i)$ is positive or
negative depending on whether or not $i$ is an integer in the Sturmian
sequence with slope $s$. We also study partial sums of the form
\begin{equation}
\label{thabs1}
\sum_{n\leq x} \frac{ a_{n+k}}{a_n}
\end{equation}
and find asymptotics for them. Next, we consider the series
\begin{equation*}
\sum_{n=1}^{\infty} \left(\frac{ a_{n+k}}{a_n}\frac{ b_{n+k}}{b_n}\right)
\end{equation*}
\noindent and show that it satisfies an identity involving $\log a$ and $\log b$ for
the case when $a$ and $b$ are irrationals. This identity bears a
superficial resemblance to a discrete analogue of Frullani's integral.
The main result of the section is deduced as a corollary from this identity.
The identity is proved by using Sturmian sequences to write the sum
\eqref{thabs1} over the positive integers, finding asymptotics for
partial sums of elements of the Beatty sequences, and using summation by
parts techniques together with some combinatorial arguments.
In the second portion of this thesis we present a Minimum Excluded
with Skipping (MES) algorithm that generalizes minimum excluded type
algorithms. This algorithm has one sequence as input and another
sequence as output. We study this algorithm in connection with Beatty,
nonhomogeneous Sturmian, and other (not necessarily quasilinear) types
of sequences. A complete characterization of this algorithm is
presented in the case of an Beatty sequence. The proofs
of these theorems use Diophantine approximation, continued fractions
and various number theoretic and combinatorial arguments.
The third portion of this thesis deals with a topic related to a
theorem that begun as a conjecture of Steinhaus. This well known Three Gap Theorem states that
there are at most three gap sizes in the sequence of fractional parts
$\{\alpha n\}_{n 
Issue Date:  20120918 
URI:  http://hdl.handle.net/2142/34298 
Rights Information:  Copyright 2012 Geremias Polanco Encarnacion 
Date Available in IDEALS:  20120918 
Date Deposited:  201208 
This item appears in the following Collection(s)

Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois