% LaTEX source code
% Last modified December 8, 1998.
% Steve Miller



\documentclass[12pt,letterpaper]{report}

\usepackage{times}
\usepackage[T1]{fontenc}
\usepackage{mathrsfs}
\usepackage{latexsym}
\usepackage[dvips]{graphics}
\usepackage{epsfig}
%\usepackage{hyperref, amsmath, amsthm, amsfonts, amscd, flafter,epsf}
\usepackage{amsmath,amsfonts,amsthm,amssymb,amscd}
\input amssym.def
\input amssym.tex


    %=======================================================
    %   THIS IS WHERE YOU PUT SHORTCUT DEFINITIONS
    %========================================================


% Note that we use a percent sign to comment out a line
% below are shortcut commands

% Theorem / Lemmas et cetera

\newtheorem{thm}{Theorem}[section]
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{exa}[thm]{Example}
\newtheorem{defi}[thm]{Definition}
\newtheorem{exe}[thm]{Exercise}
\newtheorem{rek}[thm]{Remark}
\newtheorem{que}[thm]{Question}
\newtheorem{prob}[thm]{Problem}
\newtheorem{cla}[thm]{Claim}

% shortcuts to environments

\newcommand\be{\begin{equation}}
\newcommand\ee{\end{equation}}
\newcommand\bea{\begin{eqnarray}}
\newcommand\eea{\end{eqnarray}}
\newcommand\bi{\begin{itemize}}
\newcommand\ei{\end{itemize}}
\newcommand\ben{\begin{enumerate}}
\newcommand\een{\end{enumerate}}
\newcommand\bc{\begin{center}}
\newcommand\ec{\end{center}}
\newcommand\ba{\begin{array}}
\newcommand\ea{\end{array}}
\newcommand{\tbf}[1]{\textbf{#1}}

\newcommand{\vars}[2]{ #1_1, \dots, #1_{#2} }
\newcommand{\ncr}[2]{{#1 \choose #2}}
\newcommand{\twocase}[5]{#1 \begin{cases} #2 & \text{#3}\\ #4
&\text{#5} \end{cases}   }
\newcommand{\twocaseother}[3]{#1 \begin{cases} #2 & \text{#3}\\ 0
&\text{otherwise} \end{cases}   }


% shortcut to words

\newcommand\CF{{Continued Fraction}}
\newcommand\cf{{continued fraction}}

% General Symbols

\def\notdiv{\ \mathbin{\mkern-8mu|\!\!\!\smallsetminus}}

%Blackboard Letters

\newcommand{\R}{\ensuremath{\mathbb{R}}}
\newcommand{\C}{\ensuremath{\mathbb{C}}}
\newcommand{\Z}{\ensuremath{\mathbb{Z}}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\W}{\mathbb{W}}
\newcommand{\Qoft}{\mathbb{Q}(t)}  %use in linux

% Finite Fields and Groups

\newcommand{\Fp}{ \F_p }
\newcommand{\Fpf}{ \Fp^{*} }
\newcommand{\ZnZ}{ \Z / n\Z}
\newcommand{\ZnZf}{ (\Z / n\Z)^{*}}
\newcommand{\ZqZ}{ \Z / q\Z}
\newcommand{\ZqZf}{ (\Z / q\Z)^{*}}

% Fractions

\newcommand{\fof}{\frac{1}{4}}  %oneforth
\newcommand{\foh}{\frac{1}{2}}  %onehalf
\newcommand{\fot}{\frac{1}{3}}  %onethird
\newcommand{\fop}{\frac{1}{\pi}}    %1/pi
\newcommand{\ftp}{\frac{2}{\pi}}    %2/pi
\newcommand{\fotp}{\frac{1}{2 \pi}} %1/2pi
\newcommand{\fotpi}{\frac{1}{2 \pi i}}

% Legendre Symbols

\newcommand{\jsi}[1]{ { \underline{#1} \choose p_i} }
\newcommand{\jso}[1]{ { \underline{#1} \choose p_1} }
\newcommand{\jst}[1]{ { \underline{#1} \choose p_2} }
\newcommand{\jsthree}[1]{ { \underline{#1} \choose 3} }
\newcommand\lag[2]{\ensuremath{\left(\frac{#1}{#2}\right)}}
\newcommand{\jsq}[1]{ { \underline{#1} \choose q} }            %(*/q)
\newcommand{\jsn}[1]{ { \underline{#1} \choose n} }            %(*/q)
\newcommand{\js}[1]{ { \underline{#1} \choose p} }

% Continued Fraction Shortcuts

\newcommand{\cffour}[4]{#1+\frac{1}{#2+\frac{1}{#3+\frac{1}{\ddots+\frac{1}{#4}}}}}
\newcommand{\cfthree}[3]{#1+\frac{1}{#2+\frac{1}{\ddots + \frac{1}{#3}}}}
\newcommand{\cftwo}[2]{#1+\frac{1}{\ddots + \frac{1}{#2}}    }
\newcommand{\alphampq }{ \Bigg| \alpha - \frac{p}{q} \Bigg|  }
\newcommand{\gi}[2]{ \Bigg[\frac{#1}{#2}\Bigg]  }
\newcommand{\gismall}[2]{ \Big[\frac{#1}{#2}\Big]  }


% Subscript Shortcuts for ps, qs, and as

\newcommand{\pnpo}{p_{n+1}}
\newcommand{\pmpo}{p_{m+1}}
%\newcommand{\pm}{p_m}
\newcommand{\pn}{p_n}
\newcommand{\pmmo}{p_{m-1}}
\newcommand{\pnmo}{p_{n-1}}
\newcommand{\pmmt}{p_{m-2}}
\newcommand{\pnmt}{p_{n-2}}

\newcommand{\qmpo}{q_{m+1}}
\newcommand{\qnpo}{q_{n+1}}
\newcommand{\qm}{q_m}
\newcommand{\qn}{q_n}
\newcommand{\qmmo}{q_{m-1}}
\newcommand{\qnmo}{q_{n-1}}
\newcommand{\qmmt}{q_{m-2}}
\newcommand{\qnmt}{q_{n-2}}

\newcommand{\ampo}{a_{m+1}}
\newcommand{\anpo}{a_{n+1}}
\newcommand{\am}{a_m}
\newcommand{\an}{a_n}
\newcommand{\ammo}{a_{m-1}}
\newcommand{\anmo}{a_{n-1}}
\newcommand{\ammt}{a_{m-2}}
\newcommand{\anmt}{a_{n-2}}

% matrix shortcuts

\newcommand{\mattwo}[4]
{\left(\begin{array}{cc}
                        #1  & #2   \\
                        #3 &  #4
                          \end{array}\right) }

\newcommand{\matthree}[9]
{\left(\begin{array}{ccc}
                        #1  & #2 & #3  \\
                        #4 &  #5 & #6 \\
                        #7 &  #8 & #9
                          \end{array}\right) }

\newcommand{\dettwo}[4]
{\left|\begin{array}{cc}
                        #1  & #2   \\
                        #3 &  #4
                          \end{array}\right| }

\newcommand{\detthree}[9]
{\left|\begin{array}{ccc}
                        #1  & #2 & #3  \\
                        #4 &  #5 & #6 \\
                        #7 &  #8 & #9
                          \end{array}\right| }



% greek letter shortcuts

\newcommand{\ga}{\alpha}                  %gives you a greek alpha




\begin{document}


\title{Various Tendencies of Non-Poissonian Distributions Along Subsequences 
of Certain Transcendental Numbers}

\author{Nathan Miller\thanks{E-mail: \texttt{njmiller@math.princeton.edu}} \\
\centerline{Mathematics Department} \\
 \centerline{Princeton University} \\
 \centerline{Princeton, NJ 08544}}

\date{\today}

\maketitle


\begin{abstract}
In this paper we examine the tendencies of non-Poissonian distributions that 
arise from the nearest neighbor spacings of $\{n^2\alpha\}$ for certain 
transcendental numbers. Tendencies of non-Poissonian distribution means 
the question of how the \emph{Disc} functions used depend on the 
starting $n$ and the number of $n$s used to calculate the distribution. 
For certain numbers, if we can find infinitely many
$\frac{p_n}{q_n}$s such that $|\alpha - \frac{p_n}{q_n}| < \frac{1}{q^{3+
\epsilon}_n}$, 
i.e. $\alpha$ can be approximated to order $3+\epsilon$ or greater, then 
we can show that 
the nearest neighbor spacings of $\{n^2\alpha\}$ from $n = 1,\ldots,N$
for certain values of $N$ give non-Poissonian distributions. From Roth's 
Theorem we know that these numbers must be transcendenal. If we look at
 $\{n^2\alpha\}$ for $n$ from $1$ to $q_n$ and take these in increasing order, 
calling the j-th of them $\beta_j$, we can show that the distribution of 
$\beta_{j+1} - \beta_j$ is supported on the integers and therefore is 
non-Poissonian.

The main part of this paper concerns itself with the problem of determining 
how much we can increase or decrease $N$, starting from $N = q_n$. A second 
question looked at in this paper is determining whether there will be a 
Poissonian distribution or not when $n$ goes from $K$ to $K+N-1$, where $K$ 
is a variable number and $N$ is constant throughout the tests.
\end{abstract}


\section{Introduction}

This paper presents evidence concerning the questions of the tendencies of 
non-Poissonian distributions. Tendencies of non-Poissonian 
distributions means the question of how the \emph{Disc} functions 
used depend on how many consecutive $n$s used to 
calculate the distribution and what the first $n$ is. Consider $\{n^2 \alpha
\}$ for $n=1 \to N$. Write 
these numbers in increasing order, label the $j^{th}$ of them $\zeta_j$. This 
paper is concerned with the distribution of the spacings between the 
$\zeta_j$. This is similar 
to the paper by \cite{lipschitz}, but it goes in another direction. In 
\cite{lipschitz}, he is just concerned with showing that as we increase $N$, 
the distribution tends to the 
Poissonian distribution. In this paper we look at certain irrational numbers 
which have subsequences, $N_j$, such that along this sequence the distribution 
never tends to the Poissonian distribution. These certain irrational numbers 
must be transcendental and be approximated to at least order $3+\epsilon$.


\section{Theory}

This section concerns itself with some background information about this
problem. We need to define the Poissonian distribution and what we mean when 
we say that neighbor spacings of $\{n^2\alpha\}^N_{n=1}$, where the curly 
brackets 
indicate the fractional part of the number enclosed, converge
to the Poissonian distribution. We also need to show that certain
subsequences, $N_j$, exist such that the neighbor spacings will not converge to
the Poissonian distribution for certain numbers. These subsequences are the 
main concern of this paper. The question of how much we can increase $N$, 
starting from $N=N_j$, until we return to the Poissonian 
distribution 
is experimentally analyzed. A second question of looking at the nearest 
neighbor spacings of $\{n^2\alpha\}^{K+N-1}_{n=K}$ and determining how we 
can increase $K$, while keeping $N$ constant, and get a Poissonian 
distribution is also analyzed. We will follow the route taken by \cite{TMHS}, 
essentially all of Chapter 13 (Poissonian Behavior and $\{n^k \alpha\}$), to 
derive the Poissonian distribution and prove that there exists susbsequences 
for $k^{th}$ neighbor spacings such that we get non-Poissonian distributions 
for certain numbers, when the sequences go from $n=1$ to $N$. Most of the 
times the theorems are taken verbatim from \cite{TMHS}.

\vspace{5 mm}

We first need to start off by defining what equidistribution is.

\begin{defi} We say that a sequnce $\{\alpha_k\}^\infty_{k=1}$ in [0,1] is 
\textbf{equidistributed} if for any interval [a,b]$\subset$ [0,1],

\begin{equation}
\lim_{N\to\infty}\frac{(\#\{\alpha_k \in [a,b],1\le k \le N\})}{N} = b-a
\end{equation}

\end{defi}

Both \cite{cassels} and \cite{lipschitz}, who says he follows the treatment 
of \cite{cassels}, show that for any integer $k$, $\{n^k\alpha\}$ is 
equidistributed.

The Poissonian distribution is defined as a convergence of certain induced 
probability measures. Induced probability measures make use of the 
\emph{Dirac-Delta Function}, which is an approximation to the identity 
at the origin.

\begin{defi} [Approximation to the Identity] A sequence of functions 
$g_n(x)$ is an approximation to the identity (at the origin) if

\ben
\item $g_n(x) \ge 0$
\item $\int g_n(x)dx = 1$
\item Given $\epsilon,\delta > 0$ there exists $N > 0$ such that for all $ n 
> N, \int_{|x|>\delta}g_n(x)dx < \epsilon$
\een

We can call $\delta(x)$, the Dirac-Delta function, the limit of any such 
family of $g_n(x)$
\end{defi}

If $f(x)$ is a nice function then
\begin{equation}
\int f(x)\delta(x)dx = \lim_{n\to\infty}\int f(x)g_n(x)dx = f(0)
\end{equation}

Given $N$ points $x_1,x_2,\ldots,x_N$ we can form a 
probability measure
\begin{equation}
\mu_N(x)dx = \frac{1}{N}\sum^N_{n=1}\delta(x-x_n)dx
\end{equation}

For the case of the random distribution (i.e. each $x_n$ is chosen 
independantly and uniformly in [0,1) ), $\mu_N$ converges to the Poissonian 
distribution.

Since $\int \mu_N(x)dx = 1$, and if f(x) is a nice function
\bea\label{intapprox}
\int f(x)\mu_N(x)dx &=& \frac{1}{N}\int \sum^N_{n=1}f(x)\delta(x-x_n)dx 
\nonumber \\ &=& \frac{1}{N}\sum^N_{n=1}\int f(x)\delta(x-x_n)dx  \nonumber 
\\ &=& \frac{1}{N}\sum^N_{n=1}f(x_n)
\eea

The final result in Equation \ref{intapprox} looks like a Riemann sum if the 
$x_n$s are equidistributed. In general, these $x_n$s will not be 
equidistributed, but assume that for any interval $[a,b]$, as $N \to \infty
$, the fraction of $x_n$s$(1 \le n \le N)$ in $[a,b]$ tends to $\int^b_ap(x)dx 
$ for some nice function $p(x)$:

\be\label{convergencetop}
\lim_{N\to\infty}\frac{\#\{n:1 \le n \le N, x_n \in [a,b]\}}{N} \to 
\int^b_ap(x)dx
\ee

In this case, if $f(x)$ is nice, then

\bea
\int f(x)\mu_N(x)dx &=& \frac{1}{N}\sum^N_{n=1}f(x_n) \nonumber \\ &\approx&
\sum^\infty_{k=-\infty}f(\frac{k}{N})\frac{\#\{n: 1 \le n \le N, x_n \in
[\frac{k}{N},\frac{k+1}{N}]\}}{N} \nonumber \\ &\to& \int f(x)p(x)dx
\eea

\begin{defi}[Convergence to $p(x)$] If the sequence of points $x_n$ satisfies
Equation \ref{convergencetop} for some nice function $p(x)$, we say the
probability measure $\mu_N(x)dx$ converges to $p(x)dx$
\end{defi}

We can now consider the questions of the distribution of neighbor spacings. 
Let $\alpha_n$ be a collection of points in $[0,1)$. We order them by size and
relabel them $\beta_1,\beta_2,\ldots,\beta_N$ where $\beta_k \ge \beta_l$ if 
and only if $k \ge l$

We will now look at how the differences $\beta_{j+1} - \beta_j$ are 
distributed. We will use distance on the wrapped unit interval. The distance
between $0.999$ and $0.001$ on the wrapped unit interval and in \emph{Mod 1} 
distance is $0.002$ instead of $0.998$.

\begin{defi}[mod 1 distance] Let $x,y \in [0,1)$. We define the mod $1$ 
distance from $x$ to $y$, $||x-y||$, by
\be
||x-y|| = min\{|x-y|,1-|x-y|\} 
\ee
\end{defi}

Any arithmetic operation done in this paper will always be done 
\textbf{mod 1}, and therefore it lives in $[0,1)$

The maximum value of $||x-y||$ is $\frac{1}{2}$. To show this we look at when
$|x-y| = 1-|x-y|$. This gives $|x-y| = \frac{1}{2}$ and therefore $||x-y|| = 
\frac{1}{2}$. For $|x-y| < \frac{1}{2} \to ||x-y|| = |x-y| < \frac{1}{2}$. For 
$\frac{1}{2} < |x-y| < 1 \to ||x-y|| = 1 - |x-y|  < \frac{1}{2}$. Since $|x-y| 
< 1$ the max value of $||x-y||$ must be $\frac{1}{2}$.

\begin{defi}[Neighbor Spacings] Given a sequence of numbers $\alpha_n$ in 
$[0,1)$, fix an $N$ and arrange the numbers $\alpha_n$ $(n \le N)$ in 
increasing order. Label the new sequence $\beta_j$; note that the ordering 
will depend on $N$. Let $\beta_{-j} = \beta_{N-j}$ and $\beta_{N+j} = 
\beta_j$.

\ben
\item The nearest neighbor spacings are the numbers $\beta_{j+1}-\beta_j, j 
= 1,\ldots,N$
\item The $k^{th}$-neighbor spacings are the numbers $\beta_{j+k}-\beta_j, j 
= 1,\ldots,N$
\een
\end{defi}

This paper concerns itself only with non-Poissonian distributions of 
nearest neighbor spacings, though it is best to derive everything for $k^{th}$ 
neighbor spacings.
 
Let us consider $N$ independent random variables $x_n$. Each variable is 
chosen in $[0,1)$; thus the probability that $x_n \in [a,b)$ is $b-a$. Let 
the $y_n$s be the $x_n$s arranged in increasing order. Because we have $N$ 
objects 
on the wrapped unit interval we have $N$ nearest neighbor spacings. Thus we 
expect the average spacing to be $\frac{1}{N}$.

\begin{defi}[Unfolding]Let $z_n = N y_n$. The numbers $z_n = N y_n$ have unit 
mean spacing. Thus, while we expect the average spacing between the adjacent 
$y_n$s to be $\frac{1}{N}$ units, we expect the average spacing between 
adjacent $z_n$s to be $1$ unit.
\end{defi}

On the wrapped unit interval the expected nearest neighbor spacing to be 
independent of $j$. In other words, we expect $\beta_{j+1} - \beta_j$ to have 
the same distribution as $\beta_{i+1} - \beta_i$.

When ordered in increasing order, consider $x_1 
= y_l$. What is the probability that $y_{l+k}$ is located between $\frac{t}
{N}$ and $\frac{t+\Delta t}{N}$ units to the \emph{right} of $y_l$? For this
to occur we need 
exactly $k-1$ of the $x_n$s to lie between $0$ and $\frac{t}{N}$ units to the 
right of $x_1 = y_l$, exactly one $x_n$, $y_{l+k}$, to lie between $\frac{t}
{N}$ and $\frac{t + \Delta t}{N}$ units to the right of $y_l$, and the 
remaining $x_n$s to lie at least $\frac{t + \Delta t}{N}$ units to the right 
of $y_l$. 

As the $x_n$s are independently determined, there are ${N-1 \choose k-1}$ 
choices of $x_n$ that are at most $\frac{t}{N}$ units to the right of $x_1 = 
y_l$. There are then ${(N-1)-(k-1) \choose 1}$ ways to choose the 
$x_n$ located between $\frac{t}{N}$ and $\frac{t + \Delta t}{N}$ units to the 
right of $x_1$.

Therefore we have,

\be
Prob\left(||y_{l+k} - y_l|| \in \left[\frac{t}{N},\frac{t + \Delta t}{N}
\right]\right)=\nonumber
\ee
\be= \left( \begin{array}{c}N-1\\k-1\end{array}\right) 
\left( \frac{t}{N}\right)^{k-1} \left(\begin{array}{c}(N-1)-(k-1)\\1
\end{array} \right) \frac{\Delta t}{N} \left( 1-\frac{t + \Delta t}{N} 
\right)^{N-(k+1)} \nonumber
\ee
\be  =\frac{(N-1)\cdots (N-1-(k-2))}{N^{k-1}} 
\frac{(N-1)-(k-1)}{N} \frac{t^{k-1}}{(k-1)!}\left(1-\frac{t+\Delta t}{N} 
\right)^{N-(k+1)}\Delta t \nonumber 
\ee
\be \to \frac{t^{k-1}}{(k-1)!}e^{-t}
\Delta t
\ee

We have just proven the following theorem:

\begin{thm} Consider $N$ independent random variables $x_n$ chosen from the 
uniform distribution on the wrapped interval $[0,1)$. For fixed $N$, arrange 
the $x_n$s in increasing order, labeled $y_1 \le y_2 \le \cdots \le y_N$.

From the induced probability measures from the $k^{th}$ neighbor 
spacings, as $N \to \infty$ we have
\be
\mu_{N,k}(t)dt = \frac{1}{N}\sum^N_{n=1}\delta \left(t - N\left(y_n - y_{n-k}
\right)
\right)dt \to \frac{t^{k-1}}{(k-1)!}e^{-t}dt
\ee
Using the unfolded $z_n$ with unit spacing, we have
\be
\mu_{N,k}(t)dt = \frac{1}{N}\sum^N_{n=1}\delta \left(t - \left(z_n - z_{n-k}
\right)
\right)dt \to \frac{t^{k-1}}{(k-1)!}e^{-t}dt
\ee
\end{thm}

For the nearest neighbor spacings, which are the only ones considered in this 
paper, we have
\be
\mu_{N,1}(t)dt = \frac{1}{N} \sum^N_{n=1}\delta (t - (z_n - z_{n-1}))dt \to 
e^{-t}dt
\ee

\begin{defi}[Poissonian Behavior] A sequence of points exhibits Poissonian
behavior if, in the limit as $N \to \infty$, the induced probability measures 
$\mu_{N,k}(t)dt$ converge to $\frac{t^{k-1}}{(k-1)!}e^{-t}dt$.
\end{defi}

If $\alpha$ is irrational, the sequence of points $\{n^2\alpha\}^N_{n=1}$ 
is equidistributed in $[0,1)$ and the neighbor spacings have been observed 
to be essentially Poissoniant . It is possible to show that, if $\alpha$ 
can be approximated
to order $3+\epsilon$, there is a certain subsequence of $N$ that gives 
rise to non-Poissonian behavior.

\vspace{4 mm}

If $\alpha$ can be approximated to order $3+\epsilon$ then there we can find 
infinitely many $\frac{p_n}{q_n}$ such that $|\alpha - \frac{p_n}{q_n}| < 
\frac{a_n}{q^{3+\epsilon}_n}$ where $a_n \to 0$ as $n \to \infty$

If $\beta_l = \{k^2\alpha\}$ ordered in increasing order, we can show 
that $\gamma_l = \{k^2\frac{p_n}{q_n}\}$, i.e. the ordering of $\{k^2
\frac{p_n}{q_n}\}$ is the same as the ordering of $\{k^2\alpha\}$.

\bea
k^2|\alpha-\frac{p_n}{q_n}| &=& |k^2\alpha - k^2\frac{p_n}{q_n}| \nonumber
\\ &=& ||\{k^2\alpha\} - \{k^2\frac{p_n}{q_n}\}|| \\&<& k^2 \frac{a_n}{q^3_n} 
\nonumber
\eea

For $k^2 \le q^2_n$ we get

\be
||\{k^2\alpha\} - \{k^2\frac{p_n}{q_n}\}|| < k^2\frac{p_n}{q^3_n} \le 
\frac{a_n}{q_n} < \frac{1}{2q_n}
\ee

for $n$ big enough that $a_n < \frac{1}{2}$

From this result  we can see that $\{k^2\frac{p_n}{q_n}\}$ is the closest to 
$\{k^2\alpha\}$. This implies that the orderings are the same.

\begin{thm} \label{nonpoissthm}
If $\alpha$ is irrational such that $|\alpha - \frac{p_n}
{q_n}| < \frac{a_n}{q^3_n}$ holds infinitely often, where $a_n \to 0$ as 
$n \to \infty$, there exist integers $N_j \to \infty$ such that 
$\mu_{N_j,1}(t)$ does not converge to $e^{-t}dt$
\end{thm}
\emph{Proof}: We just showed that

\be 
||\{k^2\alpha\} - \{k^2\frac{p_n}{q_n}\}|| = ||\beta_l - \gamma_l|| < 
\frac{a_n}{q_n}
\ee

If we let $N_n = q_n$ and multiply through by $N_n = q_n$ we get

\be
||N_n(\beta_l - \gamma_l)|| < a_n
\ee

We can now see that
\bea
||N_n(\beta_l - \gamma_l) - N_n(\beta_{l-1} -\gamma_{l-1})|| &<& ||N_n(\beta_l 
- \gamma_l)|| + ||N_n(\beta_{l-1} - \gamma_{l-1})|| \nonumber \\ &<& a_n + a_n 
= 2a_n
\eea

We can also see that
\bea
||N_n(\beta_l - \gamma_l) - N_n(\beta_{l-1} -\gamma_{l-1})|| &<& ||N_n(\beta_l 
- \beta_{l-1}) - N_n(\gamma_l - \gamma_{l-1})|| \nonumber \\ &<& 2a_n
\eea

Since we know that $a_n \to 0$ this resilt must mean the difference between 
$||N_n(\beta_l - \beta_{l-1})|$ and $||N_n(\gamma_l - \gamma_{l-1})||$ 
goes to zero also.

Thus, $\mu^{\alpha}_{N_n,1}(t)dt - \mu^{\frac{p_n}{q_n}}_{N_n,1}(t)dt \to 0$. 
If we look at $\mu^{\frac{p_n}{q_n}}_{N_n,1}(t)dt$ we can show that it is 
supported on the integers and therefore cannot be Poissonian.

\be
\mu^{\frac{p_n}{q_n}}_{N_n,1}(t)dt = \frac{1}{N_n}\sum^{N_n}_{l=1}\delta
\left(t - N_n(\gamma_l - \gamma_{l-1})\right)dt
\ee

We can see that

\be 
N_n(\gamma_l - \gamma_{l-1}) = N_n\gamma_l - N_n\gamma_{l-1}
\ee

We know that $\gamma_l = \{k^2\frac{p_n}{q_n}\}$,

\be
k^2\frac{p_n}{q_n} = \frac{k^2 p_n}{q_n} = \frac{Tq_n+R}{q_n} = 
T +\frac{R}{q_n}, \ee

where $T$ is an integer, $R < q_n$, and $R$ is an integer. This implies 
that 

\be
\{k^2\frac{p_n}{q_n}\} = \{T + \frac{R}{q_n}\} = \frac{R}{q_n}
\ee

Therefore,

\be
N_n\gamma_l = q_n\frac{R}{q_n} = R,
\ee
which is an integer, so $N_n(\gamma_l-\gamma_{l-1})$ is an integer and the 
induced probability measures $\mu^{\frac{p_n}{q_n}}_{N_n,1}(t)dt$ and 
$\mu^{\alpha}_{N_n,1}(t)dt$ are supported on the integers. Consequently, they 
do not converge to $e^{-t}dt$ and are not Poissonian.

The main experimental part of this paper is concerned with examining two 
different families of transcendental numbers and another specific number, all 
of which can be approximated 
to at least order $3+\epsilon$ and therefore exhibit this non-Poissonian 
behavior for certain $N_n$. The two families that are being considered are 
$\sum^{\infty}_{i=1}\frac{1}{k^{i!}}$ and $[k^{1!},k^{2!},\ldots]$, where 
the brackets denote the continued fraction of the numbers enclosed. The 
specific number being examined is the number labeled $\eta$ in 
\cite{lipschitz} and constructed in \cite{RSZ}.

\section{Experimental Numbers}

In this section we prove that the two families used in the experiments can be 
approximated arbitrarily well. The number $\eta$ will also be constructed and 
shown to be of order $3+\epsilon$.

We will first show that $\sum^{\infty}_{i=1}\frac{1}{k^{i!}}$ can be 
approximated abritrarily well. This construction is also essentially done in 
\cite{TMHS}, in Chapter 12 Section 3 (Constucting Transcendental Numbers), 
though it is a common construction and it is modified from \cite{TMHS} to show 
that $k$ can be any integer, not just $k=10$.. 
If we let $\frac{p_n}{q_n} = \sum^n_{i=1} 
\frac{1}{k^{i!}}$, then $q_n = \frac{1}{k^{n!}}$ and

\bea
|\alpha - \frac{p_n}{q_n}| = |\sum^{\infty}_{i=1}\frac{1}{k^{i!}} - 
\sum^{n}_{i=1}\frac{1}{k^{i!}}| = \sum^{\infty}_{i=n+1}\frac{1}{k^{i!}} 
\nonumber \\ = \frac{1}{k^{(n+1)!}} \left(1+\frac{1}{k^{n+2}}+
\frac{1}{k^{(n+2)(n+3)}}+ \cdots \right) \nonumber \\ < \frac{2}{k^{(n+1)!}}
 = \frac{2}{(k^{n!})^{n+1}} < \frac{2}{q^{n+1}_n} \le \frac{2}{q^N_n}
\eea

Since we can choose $N$ arbitrarily, we can approximate it arbitrarily well. 
We will call this number $\beta_k$ from now on, where the $k$ refers to the 
$k$ in the infinite sum.

\vspace{6 mm}

To show that $\alpha=[k^{1!},k^{2!},\ldots]=[a_0,a_1,\ldots]$ can be 
approximated arbitrarily well, we will again follow a construction in 
\cite{TMHS}, also in Chapter 12 Section 3 (Constructing Transcendental 
Numbers), though it is again a common construction and again it is modified 
from \cite{TMHS} to show it is true for all integers, $k$, not just 
$k=10$.
Let $\frac{p_n}{q_n} = [k^{1!},k^{2!},\ldots,k^{n!}]$. Then $a_n = k^{(n+1)!}
$ and

\be\label{contfrac1}
|\alpha - \frac{p_n}{q_n}| = \frac{1}{q_n q_{n+1}'} = \frac{1}{q_n 
(a_{n+1}'q_n + q_{n-1})} < \frac{1}{a_{n+1}} = \frac{1}{k^{(n+1)!}}
\ee

Since $q_k = a_kq_{k-1}+q_{k-2}$ we see that

\be
\frac{q_{k+1}}{q_k} = a_{k+1} + \frac{q_{k-1}}{q_k} < a_{k+1} + 1
\ee

Using this inequality for $q_n$ we get

\bea\label{contfrac2}
q_n = q_0 \frac{q_1}{q_0}\frac{q_2}{q_1}\cdots\frac{q_n}{q_{n-1}} &<& 
(a_0+1)(a_q+1)\cdots(a_n+1) \nonumber \\ &=& (1+\frac{1}{a_0})(1+\frac{1}
{a_1})\cdots (1+\frac{1}{a_n})a_1a_2\cdots a_n \nonumber \\ &<& 2^na_0a_1
\cdots a_n = 2^n k^{(1!+\cdots+(n+1)!)} \nonumber \\ &<& k^{2(n+1)!} = a^2_n
\eea

Combining Equations \ref{contfrac1} and \ref{contfrac2} we get

\bea
|\alpha - \frac{p_n}{q_n}| &<& \frac{1}{a_{n+1}} = \frac{1}{a^{n+1}_n} 
\nonumber \\ &<& \left(\frac{1}{a^2_n}\right)^{\frac{n}{2}} < \left(\frac
{1}{q_n}\right)^{\frac{n}{2}} = \frac{1}{q^{\frac{n}{2}}_n}
\eea

Therefore this $\alpha$ can be approximated arbitrarily well because all 
$\frac{p_n}{q_n}$ with $n>2*t$ will be approximated to at least order $t$. 
We will call this well-approximable number $\gamma_k$ from now on, where 
the $k$ refers to the $k$ in the continued fraction representation of the 
number.

\vspace{6 mm}

The construction of $\eta$ used here is the same as in \cite{RSZ} and used in 
\cite{lipschitz}. To start off we will define a pair of integers $(r_m,v_m)$ 
by $r_{-1} = v_{-1} = 0, r_0 = v_0 = 1, r_1 = v_1 = 1$ and for $m \ge 1$

\bea
v_{m+1} &=& r_m v^2_m + v_{m-1} \nonumber
\\ r_{m+1} &=& [log_2 (v_{m+1})]
\eea
where the brackets mean the greatest integer less than or equal to the value 
enclosed. Now we set $a_0 = 1$, and for $m \ge 0$

\be
a_{m+1} = r^2_m v^2_m + 2r_m v_{m-1}
\ee

We let $\eta$ be $[a_0,a_1,\ldots] = [1,1,3,6,\ldots]$ To show that this 
number is of type 
$3+\epsilon$ we need to show that $q_m = v^2_m$. To do this we look at the 
recursion relation for $q_{m+1} = a_{m+1}q_m + q_{m-1}$ and use induction

\bea
q_{m+1} &=& a_{m+1} v^2_m + v^2_{m-1} = (r^2_m v^2_m + 2 r_m v_{m-1}) v^2_m + 
v^2_{m-1} \nonumber \\ &=& (r_m v^2_m + v_{m-1})^2 = v^2_{m+1}
\eea
and $q_{m+1}=v^2_{m+1}$ as long as $q_m=v^2_m$.
Since $q_0 = v^2_0 = 1$ and we can define $q_{-1} = v^2_{-1} = 0$ it is 
proved. From the previous equation we can see

\be 
q_{m+1} \sim r^2_m q^2_m \sim q^2_m (log (q_m))^2
\ee

From basic continued fractions we have

\be
|\eta - \frac{p_n}{q_n}| < \frac{1}{q_n q_{n+1}} \sim \frac{1}{q^3_n (log 
(q_n))^2}
\ee

And thus $\eta$ is of order $3+\epsilon$

\section{Experimental Data}

In this section we will discuss the experimental data and what can be 
determined from this data about the tendencies of non-Poissonian 
distributions for 
nearest neighbor spacings. It is best first to describe the experiments that 
were done and how they were carried out. As was shown in Theorem 
\ref{nonpoissthm}, if 
the number of points $N=q_N$, a non-Possonian distribution supported 
on the integers arises. The question of how many additional points does it 
take, increasing N, to return to the Poissonian distribution, and 
how many points can we take away before we return again to the Poissonian 
distribution, is investigated. Througout our discussion of the first question, 
the sequence $\{n^2\alpha\}$ starts at $n=1$. It is a separate questiuon 
whether the distribution of nearest neighbor spacings is different when 
a sequence of a defined length starts at some different $n$ such as $10^8$ 
or maybe a $q_n$.

All these experiments were done using \emph{Mathematica}. The experiments were 
made up of the following steps:

\ben
\item Calculate the numbers $\{n^2\alpha\}$ for $n=n_{min},\ldots, n_{max}$
\item Sort all the numbers
\item Calculate all the nearest neighbor spacings and make them unit mean 
spacing by multiplying by $n_{max} - n_{min} + 1$.
\item Sort the nearest neighbor spacings
\item Calculate \emph{Disc} and \emph{Disc2}, the two discrepancy functions 
used, defined below
\een

\emph{Disc} is the same discrepancy function used in \cite{lipschitz} to 
calculate the largest difference between the experimental cumulative 
probability distribution (the probability that the difference is less than a 
certain value, t) and the Poissonian cumulative probability distribtion, 
$\int^t_0 e^{-x}dx = 1 - e^{-t}$. \emph{Disc2} calculates the average 
difference between the distributions.

\vspace{6 mm}

To answer the first question about how far away from $N=q_n$ do we need to 
go to return to the 
Poissonian distribution it is best first to see exactly how the numbers 
behave at $N=q_n$

\vspace{6 mm}

\begin{tabular}{|r|r|r|r|} \hline 
$\beta_k$ & $q_n$ & Disc & Disc2
\\ \hline k=2 & 64 & 0.8016 & 0.3594 
\\ k=3 & 729 & 0.6228 & 0.2338
\\ k=4 & 4096 & 0.8328 & 0.3598
\\ k=5 & 15625 & 0.5833 & 0.2124
\\ k=6 & 46656 & 0.9295 & 0.4343
\\ k=7 & 117649 & 0.5625 & 0.1829
\\ k=8 & 262144 & 0.8333 & 0.3598
\\ \hline $\gamma_k$ & $q_n$ & Disc & Disc2
\\  \hline k=2 & 257 & 0.4981 & 0.1525
\\ k=3 & 6562 & 0.7339 & 0.2893
\\ k=4 & 65537 & 0.5000 & 0.1513
\\ k=5 & 390626 & 0.7353 & 0.2900
\\ k=6 & 1679617 & 0.7353 & 0.2899
\\  \hline $\eta$ & 2704 & 0.8614 & 0.3940
\\ \hline $e$ & 8544 & 0.0123 & 0.0041
\\ $e$ & 208524 & 0.0029 & 0.0011
\\ \hline
\end{tabular} 

\vspace{6 mm}

Looking at this table, we cannot make any specific guesses as to what the 
\emph{Disc} and \emph{Disc2} would be for a random transcendental number 
that satisfied the 
conditions of Theorem \ref{nonpoissthm} at a random $N=q_n$. For $\beta_k$ it 
appeared that, as $k$ is increased, the $Disc$ alternates back and forth 
between values around 0.8 and around 0.6. However, for $\gamma_k$ it looks 
like it alternates 
between 0.73 and 0.5 until we go from $k=5$ to $k=6$, where the the 
\emph{Disc} and \emph{Disc2} stay essentially the same. $\eta$ seems to 
have a somewhat high \emph{Disc} and \emph{Disc2}, only topped by $\beta_6$, 
but that means little since there is no strong correlation between $N=q_n$ and 
the discrepancy. The base of the natural logarithm, $e$, is in this table to 
show that the big discrepancy observed for the other numbers is 
not normal for all numbers. More information about the discrepancies 
of trascendental numbers not of order $3+\epsilon$ and of algebraic numbers 
can be found in \cite{lipschitz}. 

\vspace{6 mm}

The following graph is a plot of the cumulative probability distributions for
$\beta_5$ where $N=q_n=15625$. This plot has a \emph{Disc} $=0.5833$ and a 
\emph{Disc2} $=0.2124$

\begin{center}
\scalebox{1.35}[1.35]{\includegraphics{cumprob5beta_gr1.eps}}
\end{center}
\begin{center}
Plot 1
\end{center}

\vspace{4 mm}

In this graph, the experimental probability is the red line and the Poissonian
theoretical probability is the blue line. The experimental probability is 
supported on the integers, although the height at each integer roughly 
follows the 
theoretical probability.

In contrast to a $Disc$ of $0.5833$, the next plot is a plot of $N=q_n=2704$ 
for the number $\eta$.

\begin{center}
\scalebox{1.3}[1.3]{\includegraphics{cumprob_eta_gr1.eps}}
\end{center}
\begin{center}
Plot 2
\end{center}

\vspace{4 mm}

This plot has a $Disc$ of $0.8614$ and a $Disc2$ of $0.3940$. $\eta$ is a 
different kind of number than $\beta_5$, and it is obvious from this 
plot. This plot has more different integers on which the experimental 
curve is supported; it also follows the theoretical plot less closely. 

One thing that should be taken from these plots is the origin of the value of 
\emph{Disc}. For all the plots with \emph{Disc} $\ge 0.4$ or so 
the value comes from the points supported on the integer $0$. \emph{Disc} is 
the 
probability that the we have a difference $< 0.5$. Since the differences 
are supported on the integers it is also the probability we have a 
difference $< 0.4$, $< 0.2$, or $<0.05$. The plots given above are at the 
extremes when 
it comes to \emph{Disc} and \emph{Disc2} when evaluated at $N=q_n$. If one 
looks at the plots for all 
the numbers given in the table above, he will find that the cirves do roughly 
follow the Poissonian distribution, but they are supported on the integers 
and have a high value supported at $0$.

\vspace{4 mm}

After looking at the how the distributions behave at $N=q_n$ it is now time to 
look at how the distributions behave away from $N=q_n$. The following two 
graphs are 
plots of the values of \emph{Disc} and \emph{Disc2} at different values of 
$N$. The x-axis is $Log_{10}(x)$ and the y-axis is the value of the 
\emph{Disc}s. The vertical line is where $x = q_n$; the red line is the 
value of \emph{Disc}, and the blue line is the value of \emph{Disc2}. Each 
plot was created using an interpolation of order 3 to connect between 25 and 
30 points run over the range of the plot. Plot 3 is a plot of $\beta_4$ and 
Plot 4 is a plot of $\eta$. They were chosen because it was possible in these 
cases to get points 
sufficiently above and below $N=q_n$ such that their distributions were 
essentially Poissonian. 
For almost every other number it was only poissible to get a good plot 
either above or below $N=q_n$, but not both.

\begin{center}
\scalebox{1.35}[1.35]{\includegraphics{discplot4beta_gr1.eps}}

Plot 3
\end{center}

\vspace{4 mm}

\begin{center}
\scalebox{1.35}[1.35]{\includegraphics{discploteta_gr1.eps}}

Plot 4
\end{center}

\vspace{4 mm}

The first thing to mention is a comment relating to \cite{lipschitz}. In his 
paper he mentions that it took him up to $N=50000$ until $\eta$ started 
behaving as he felt it should be behaving, Poissonian-like. If he had looked 
at $N=200 \to 300$ he would have noticed that it was Poissonian for these 
small $N$s. It took that long for Lipschitz to notice Poissonian behavior, 
because $\eta$ satisifes the requirements of Theorem \ref{nonpoissthm} and he 
was in the neighborhood of $N=q_n$. Consequntly his graphs felt the effects 
of the non-Poissonian, integer-supported distribution, and only at $N=50000$ 
for a $q_n = 2704$, did the graph return to Poissonian behavior. With a 
computer with sufficient memory and able to do $N=q_5=182,925,625$, he would 
have noticed that the graph again became non-Poissonian and integer-supported.

\vspace{5 mm}

To get on with looking at the graphs above, the graph of $\beta_4$ has a 
$q_3=4096$ and the graph of $\eta$ has a 
$q_4=2704$. The next $q_n$ for $\beta_4$, $q_4$, is $281474976710656$ and the 
next $q_n$ for $\eta$, $q_5$, is as previously mentioned $182925625$. The 
first $q_n$s listed have similar values, both on the order of $10^3$; 
however the next $q_n$s are very different. The $q_n$ for $\beta_4$ is on the 
order of $10^{14}$, whereas that for $\eta$ is on the order of $10^8$. This 
is what causes the differences in the two graphs. The values for which the 
distributions seem to return to Poissonian behavior, above or below $N=q_n$, 
depend on the magnitude of $q_n$ and the magnitude of either $q_{n-1}$ or 
$q_{n+1}$ depending on whether we are looking at how much $N$ mus be increased 
or decreased to return to Poissonian behavior. The following is a table 
of discrepancies for different values of $N$ for different numbers above the 
$q_n$ for each number. Each $N$ 
is either the the value with which \emph{Disc} $\approx 0.1$ and 
\emph{Disc2} $\approx 0.05$ or the value at $1,000,000$. $1,000,000$ is 
essentially the largest $N$ possible for the machine to run the program in 
a reasonable amount of time and not run out of memory. The program 
halted because of lack of memory when it was run with $N = 10,000,000$.

\vspace{4 mm}
\begin{tabular}{|r|r|r|r|r|r|r|}
\hline $\alpha$ & $q_n$ & $\approx q_{n+1}$ & $N$ & \emph{Disc} & \emph{Disc2} 
& $N/q_n$
\\ \hline $\beta_2$ & $64$ & $1.7 * 10^7$ & $1400$ & $0.0884$ & $0.0480$ 
& $21.88$
\\ $\beta_3$ & $729$ & $2.8 * 10^{11}$ & $45000$ & $0.0931$ & $0.0503$
& $61.73$
\\ $\beta_4$ & $4096$ & $2.8 * 10^{14}$ & $1000000$ & $0.0905$ & $0.0492$
& $244.14$
\\ $\beta_5$ & $15625$ & $5.9 * 10^{16}$ & $1000000$ & $0.6503$ & $0.3962$
& N$\backslash$A
\\ $\gamma_2$ & $257$ & $4.3 * 10^9$ & $105000$ & $0.0935$ & $0.0614$
& $408.56$
\\ $\gamma_3$ & $6562$ & $1.9*10^{15}$ & $1000000$ & $0.9972$ & $0.4981$
& N$\backslash$A
\\ $\eta$ & $2704$ & $1.8 * 10^8$ & $50000$ & $0.0932$ & $0.0533$
& $18.49$
\\ \hline 
\end{tabular}

\vspace{6 mm}

One thing to notice from this table concerns the entries for $\beta_3$ and 
$\gamma_2$. $\beta_3$ has a higher $q_n$ and a higher $q_{n+1}$ than 
$\gamma_2$, yet the value of $N$ that gives a \emph{Disc} $\approx 
0.1$ and a \emph{Disc2} $\approx 0.05$ is smaller for $\beta_3$ than 
$\gamma_2$, henceforth the value of $N/q_n$ for $\beta_3$ is much smaller than 
for $\gamma_2$. Also, $\eta$ has a higher $q_n$ by a factor of $\approx 5$ and 
a lower $q_{n+1}$ by a factor of $\approx 10^3$ than does $\beta_3$, yet it 
requires approximately the same value of $N$ to give the \emph{Disc}s listed. 
There is clearly something 
more going on than is implied by a relation between the $q_n$s that determines 
how far away from $N=q_n$ we must go to return to Poissonian behavior. There 
is not common value of $N/q_n$ from which we can predict a value of $N$ where 
we will get a Poissonian distribution. The 
tendency to return to a Poissonian distribution probably also 
depends on what the value of $t$ in the following equation:

\be
|\alpha - \frac{p_n}{q_n}| < \frac{1}{q^t_n}
\ee

$\beta_k$, $\gamma_k$, and $\eta$ all have different values of t, which 
might explain the differences in the $N$s in the table above. However, 
exploring this point any further is beyond the scope of this present work..

\vspace{5 mm}

The previous tables and graphs show only how the \emph{Disc}s behave around 
one $q_n$ which is on the order of $10^3$ or $10^4$. Only $\beta_2$ gives 
a small indication at what happens at a second $q_n$. $\beta_2$ has $q_3=64$ 
and $q_4=16777216$. The following is a table of discrepancies for a couple 
values of $N$ for $\beta_2$

\vspace{4 mm}

\begin{tabular}{|r|r|r|}
\hline $N$ & \emph{Disc} & \emph{Disc2}
\\ \hline $64$ & $0.8016$ & $0.3594$
\\ $50000$ & $0.0102$ & $0.0027$
\\ $1000000$ & $0.0845$ & $0.0243$
\\ $5000000$ & $0.4408$ & $0.1535$
\\ \hline
\end{tabular}

\vspace{4 mm}

One can see from the discrepancies that the distribution is non-Poissonian at 
$N=q_3=64$, then is essentially Poissonian for $N$ between $10^4$ and $10^6$. 
Finally as it approaches $N=q_4=16777216$, the distribution is non-Poissonian 
again. This is the only experimental proof that Theorem \ref{nonpoissthm} is 
true for $q_n \ge 1000000$. All the experimental proof obtained in this paper 
is for single $q_n$s for a number with $q_n \approx 10^3 \to 10^4$. In the 
future, with a more powerful computer, it might be instructive to look at how 
the distributions behave for several $q_n$s for a single number.

\vspace{6 mm}

The next experimental results concern different starting values for $n$ in 
$\{n^2\alpha\}$. The discrepancies for the same size blocks (i.e. number of 
numbers in the sequence), but with different 
starting positions, were compared. In the previous experiments, the starting 
positions were always $n=1$. Now we are looking at starting positions of 
$n=10^8$, $n=q_n$, or something similar.

The following is a table of values of the discrepancies for a block of size 
$2704$, with $\eta$ as our number. A block size of $2704$ was chosen because 
it is a $q_n$ of 
$\eta$. The input value that changes is the minimum $n$

\vspace{3 mm}
\begin{tabular}{|r|r|r|}
\hline Min $n$ & \emph{Disc} & \emph{Disc2}
\\ \hline $1$ & $0.8614$ & $0.3940$ 
\\ $2704$ & $0.8332$ & $0.3860$
\\ $182925625$ & $0.8611$ & $0.3937$
\\ $q_6$ & $0.8611$ & $0.3937$
\\ $2704 + 10^4$ & $0.7403$ & $0.3604$
\\ $182925625 + 10^4$ & $0.7632$ & $0.3673$
\\ $q_6 + 10^4$ & $0.7632$ & $0.3673$
\\ $2704 + 10^5$ & $0.3312$ & $0.1833$
\\ $182925625 + 10^5$ & $0.3397$ & $0.1871$
\\ $q_6 + 10^5$ & $0.3397$ & $0.1871$
\\ $2704 + 10^6$ & $0.0173$ & $0.0049$
\\ $182925625 + 10^6$ & $0.0149$ & $0.0044$
\\ $q_6 + 10^6$ & $0.0149$ & $0.0044$
\\ $182925625 - 10^5$ & $0.3473$ & $0.1909$
\\ $q_6 - 10^5$ & $0.3473$ & $0.1909$
\\ \hline
\end{tabular}

\vspace{5 mm}
$2704=q_4$, $182925625=q_5$, and $q_6=5655041790912713329$. The values of 
the discrepancies are essentially the same at values of Min $n$ that are the 
same distance from a $q_n$. 
For $q_5$ and $q_6$ they are the same to $4$ significant figures. The reason 
$q_4$ is different is probably because it is closer to $q_3$, $q_2$, 
and $q_1$ than the increase required of the Min $n$ to return to the 
Poissonian distribution and there is probably some influence on the 
distribution from these other $q_n$s. Their return to a Poissonian 
distribution 
acts differently from looking at different size blocks with the same Min $n$. 
In this case there is a certain distance from $q_n$ at which the distribution 
returns to Poissonian. In the first case we looked at, variable block size and 
constant Min n, it look longer to return to 
Poissonian distribution when $q_n$ was larger. To show that the results from 
this case, constant block size and variable Min n, are not a fluke 
the following is a table for $\alpha = \beta_5$, block size of $q_3=15625$.

\vspace{4 mm}

\begin{tabular}{|r|r|r|}
\hline Min $n$ & \emph{Disc} & \emph{Disc2}
\\ \hline $q_3$ & $0.5831$ & $0.2123$
\\ $q_4$ & $0.5832$ & $0.2124$
\\ $q_3 + 10^6$ & $0.5749$ & $0.2104$
\\ $q_4 + 10^6$ & $0.5751$ & $0.2104$
\\ $q_3 + 5*10^7$ & $0.2471$ & $0.1290$
\\ $q_4 + 5*10^7$ & $0.2472$ & $0.1290$
\\ $q_3 + 3*10^8$ & $0.1006$ & $0.0462$
\\ $q_4 + 3*10^8$ & $0.1006$ & $0.0462$
\\ \hline
\end{tabular}

\vspace{5 mm}

In this table $q_3=15625$ as was mentioned, and $q_4=59604644775390625$. The 
distance required to return to the Poissonian distribution is different 
than that for $\eta$, but as was also the case for $\eta$, the same distance 
from a $q_n$ gives the same discrepancies. If we had chosen any other 
$\alpha$, we would have the \emph{Disc}s act the same. This is essentially 
all that we can deduce about the tendencies of non-Poissonian distributions 
for certain transcendental numbers.

\section{Conclusion and What Comes Next}

Nothing concrete can be deduced from the data taken. What could be seen has 
already 
been mentioned in the previous section. It was quite interesting to see that 
for the sequences of $n=1 \to N$, if 
$N=q_n+R$, that for a larger $q_n$, a larger $R$ was needed to have a 
distribution that looked like the Poissonian distribution. This was quite 
different that if we looked at $n=q_n + R \to q_n + R + N - 1$. In this case 
for all different $q_n$s of the same $\alpha$, the discrepancies were the same 
if $R$ is held constant. 

\vspace{4 mm}

As was metioned at the beginning of the Experimental Data section, all the 
experiments were done using \emph{Mathematica}. The prograns, whose outline is 
also given in that section, were originally just the same as the \emph{Maple} 
programs used to get the data in \cite{lipschitz}, except changed to run in 
\emph{Mathematica}, but the programs were eventually modified when 
needed to change the range of $n$. The only further experiments for the 
nearest neighbor distributions that could be 
done would be to look at larger $q_n$s than the ones done here. This 
is impossible until there are much faster computers, with more RAM. It could 
take anywhere from 3 to 30 minutes to calculate the distribution for 
$n=1 \to 1,000,000$. It would take even longer when looking at $n=q_n + R \to 
q_n + R + N - 1$ because more digits of $\alpha$ was needed. When starting at 
$n=1$, $30$ digits of $\alpha$ was used. When starting from $\alpha=q_n + R$, 
the numbers of digits used was based on the highest value of $n$ used such 
that there would be $\approx 10$ digits to the right of the decimal place when 
looking at $n^2 \alpha$.

\vspace{4 mm}

Because right now it is almost impossible to get any data for higher $q_n$s 
for nearest neighbor distributions, the only reasonable things to do next 
would be to look other $k^{th}$ neighbor distributions and see if the results 
are similar to the results for nearest neighbor distributions. Only nearest 
neighbor distributions were considered in this paper and it would be 
interesting to see if in the higher neighbor distributions it took longer 
to return to a Poissonian distribution or it took around the same amount of 
increase on $N$ when starting at $n=1$ and also when going from $n=q_n+K$ to 
$n=q_n+K+N-1$ (i.e. the second question that was begin answered in the 
paper), do we get the same \emph{Disc}s for the same $K$s as the nearest 
neighbor distribution. It would not be that hard to modify the programs I used 
to make them run for $k^{th}$ neighbor distributions. The parts that would 
need to be changed would be the actual calculation of the neighbor spacings, 
changed from $1^{st}$ neighbor spacings to general $k^{th}$ neighbor 
spacings and the theoretical Poissonian distribution from 
$\int^x_0 e^{-t} dt$ to $\int^x_0 \frac{t^{k-1}}{(k-1)!} e^{-t}dt$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{thebibliography}{99}

\bibitem[Ca]{cassels}
\newblock J. W. S. Cassels, \emph{An Introduction to Diophantine 
Approximations}, Vol II. Second Edition. John Wiley $\&$ Sons, Inc., 
New York-London-Sydney 1971

\bibitem[Li]{lipschitz}
\newblock R. Lipshitz, \emph{Numerical Results Concerning the 
Distribution of \(n^2\alpha\)}, Junior Research Project 2000-2001

\bibitem[RSZ]{RSZ}
\newblock Ze\'ev Rudnick, P. Sarnak, and A. Zaharescu, \emph{The
Distribution of Spacings Between the Fractional Parts of
\(n^2\alpha\)}, Invent. Math. $145$ ($2001$), no. $1$, $37-57$.

\bibitem[TMHS]{TMHS}
\newblock R. Takloo-Bighash, S. J. Miller, H. Helfgott, F. Spinu, 
\emph{Junior Research Seminar: Diophantine Analysis and Approximations}
Lecture Notes, $2002$

\end{thebibliography}



\end{document}
