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\begin{document}
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\begin{center}
\vskip 1cm{\LARGE\bf
Dedekind Sums with Arguments near Certain \\
\vskip .1in
Transcendental Numbers
}
\vskip 1cm
\large
Kurt Girstmair\\
Institut f\"ur Mathematik \\
Universit\"at Innsbruck \\
Technikerstr.\ 13/7 \\
A-6020 Innsbruck \\
Austria \\
\href{mailto:Kurt.Girstmair@uibk.ac.at}{\tt Kurt.Girstmair@uibk.ac.at}
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\begin{abstract}
We study the asymptotic behavior of the classical Dedekind sums $s(s_k/t_k)$ for the sequence of convergents $s_k/t_k$
$k\ge 0$, of the transcendental number
\BD
\sum_{j=0}^\infty\frac 1{b^{2^j}},\ b\ge 3.
\ED
In particular, we show that there are infinitely many open intervals of constant length such that the
sequence $s(s_k/t_k)$ has infinitely many transcendental cluster points in each interval.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction and result}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\DED sums have quite a number of interesting applications in
analytic number theory (modular forms), algebraic number theory (class numbers),
lattice point problems and algebraic geometry
(for instance \cite{{Ap}, {Me}, {RaGr}, {Ur}}).
Let $n$ be a positive integer and $m\in \Z$, $(m,n)=1$. The classical \DED sum $s(m/n)$ is defined by
\BD
s(m/n)=\sum_{k=1}^n ((k/n))((mk/n))
\ED
where $((\cdots))$ is the usual sawtooth function (for example, \cite[p.\ 1]{RaGr}).
In the present setting it is more
natural to work with
\BD
S(m/n)=12s(m/n)
\ED instead.
In the previous paper \cite{Gi} we used the Barkan-Hickerson-Knuth-formula to study the asymptotic behavior
of $S(s_k/t_k)$ for the convergents $s_k/t_k$ of transcendental numbers like $e$ or $e^2$.
In this situation the limiting behavior of $S(s_k/t_k)$ was fairly simple. It is much more complicated, however,
for the transcendental number
\BE
\label{0.2}
x(b)=\sum_{j=0}^\infty\frac 1{b^{2^j}},\ b\ge 3.
\EE
In fact, we have no full description of what happens in this case. Its complexity is illustrated by the following
theorem, which forms the main result of this paper.
\begin{theorem} % Theorem 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{t1}
Let $s_k/t_k$, $k\ge 0$, be the sequence of convergents of the number $x(b)$ of {\rm (\ref{0.2})}.
Then the sequence $S(s_k/t_k)$, $k\ge 0$, has infinitely many transcendental cluster points
in each of the intervals
\BD
\left(b-10-2i+\frac 1b,b-9-2i+\frac 1{b-1}\right),\ i\ge 0.
\ED
\end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Note that each of the intervals of Theorem \ref{t1} has the length $1+1/(b(b-1))$, whereas the distance
between two neighboring intervals is $1-1/(b(b-1))$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The integer part}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We start with the continued fraction expansion $[a_0,a_1,a_2,\LD]$ of an arbitrary irrational number $x$.
The numerators and denominators of its convergents
\BE
\label{2.0.1}
s_k/t_k=[a_0,a_1,\LD,a_k]
\EE
are defined by the recursion formulas
\begin{eqnarray}
\label{2.0}
s_{-2}=0,&& s_{-1}=1,\hspace{5mm} s_k=a_ks_{k-1}+s_{k-2} \,\MB{ and } \nonumber\\
t_{-2}=1,&& t_{-1}=0,\hspace{5mm} t_k=a_kt_{k-1}+t_{k-2},\,\MB{ for } k\ge 0.
\end{eqnarray}
Henceforth we will assume $00$, $q'\ge 0$,
we obtain
\BD
x+\frac{p(x+1)+p'x}{q(x+1)+q'x}=\alpha.
\ED
This, however, means that $x$ satisfies a quadratic equation over the field $\Q(\alpha)$.
Accordingly, $x$ is algebraic, a contradiction.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of Theorem \ref{t1}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As in the setting of Proposition \ref{p1}, let $i\ge 0$ and $r\ge 1$ be given and $n_l=2^{2+l}$, $l=1,\LD, r$.
Suppose that the numbers $k_{i,l}$ are defined as in {\rm (\ref{2.12})}. Let $\WH n$ be a power
of $2$, $\WH n\ge 2^{i+r+3}$. By Proposition \ref{p1},
\BD
L(\WH n+k_{i,l})=b-7-2i.
\ED
If $\WH n$ tends to infinity, Proposition \ref{p2} says that $t_{\WH n+k_{i,l}}/t_{\WH n+k_{i,l}-1}$
tends to
\BD
t(k_{i,l})=[a_{k_{i,l}}, a_{k_{i,l}-1},\LD,a_2,b-2,(x+1)/x].
\ED
Therefore $t_{\WH n+k_{i,l}-1}/t_{\WH n+k_{i,l}}$ tends to $1/t(k_{i,l})$. Altogether, we have
\BD
S(s_{\WH n+k_{i,l}}/t_{\WH n+k_{i,l}})\to b-10-2i+x+\frac{1}{t(k_{i,l})}.
\ED
For $l