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\begin{center}
\vskip 1cm{\Large \bf
A Note on the Lonely Runner Conjecture} \\
\vskip 1cm
\large Ram Krishna Pandey \\
D\'{e}partement de Math\'{e}matiques\\
Universit\'{e} Jean Monnet \\
23, rue Dr.\ Paul Michelon \\
42023 Saint-Etienne\\
France \\
\href{mailto:ram.krishna.pandey@univ-st-etienne.fr}{\tt ram.krishna.pandey@univ-st-etienne.fr}
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\begin{abstract}
Suppose $n$ runners having nonzero distinct constant speeds run laps
on a unit-length circular track. The \emph{Lonely Runner Conjecture}
states that there is a time at which all the $n$ runners are
simultaneously at least $1/(n+1)$ units from their common starting
point. The conjecture has been already settled up to six ($n \leq
6$) runners and it is open for seven or more runners. In this paper
the conjecture has been proved for two or more runners provided the
speed of the $(i+1)$th runner is more than double the speed of the
$i$th runner for each $i$, arranged in increasing order.
\end{abstract}
\section{Introduction and Summary}
The conjecture in its original form stated by Wills \cite{wills} and
also independently by Cusick \cite{cusick} is as follows:
\begin{quote}
For any $n$ positive integers $w_1,w_2, \ldots, w_n,$ there is a
real number $x$ such that
\[\|w_ix\| \geq \frac{1}{n+1},\]
for each $i=1,2, \ldots, n$, where for a real number $x$, $\|x\|$ is
the distance of $x$ from the nearest integer.
\end{quote}
Due to the interpretation by Goddyn \cite{bienia}, the conjecture is
now known as the ``Lonely Runner Conjecture".
\begin{quote}
Suppose $n$ runners having nonzero distinct constant speeds run laps
on a unit-length circular track. Then there is a time at which all
the $n$ runners are simultaneously at least $1/(n+1)$ units from
their common starting point.
\end{quote}
The term ``lonely runner'' reflects an equivalent formulation in
which there are $n+1$ runners with distinct speeds.
\begin{quote}
Suppose $n+1$ runners having nonzero distinct constant speeds run
laps on a unit-length circular track. A runner is called lonely if
the distance (on the circular track) between him (or her) and every
other runner is at least $1/(n+1).$ The conjecture is equivalent to
asserting that for each runner there is a time when he (or she) is
lonely.
\end{quote}
The case $n=2$ is very simple. For $n = 3,$ Betke and Wills
\cite{betkewills} settled the conjecture while Wills was dealing
with some Diophantine approximation problem and also independently
by Cusick \cite{cusick} while Cusick was considering $n$-dimensional
``view-obstruction'' problem. The case $n = 4$ was first proved by
Cusick and Pomerence \cite{cusickpom} with a proof that requires a
work of electronic case checking. Later, Bienia et al. \cite{bienia}
gave a simpler proof for $n = 4$. The case $n = 5$ was proved by
Bohman, Holzman and Kleitman \cite{bohman}. A simpler proof for the
case $n=5$ was given by Renault \cite{renault}. Recently, Barajas
and Serra (\cite{serra}, \cite{barserra}) proved the conjecture for
$n=6.$ Goddyn and Wong \cite{goddyn} gave some tight instances of
the lonely runner. For $n \geq 7$ the conjecture is still open. We
prove the conjecture for two or more runners provided the speed of
the $(i+1)$th runner is more than double the speed of the $i$th
runner for each $i$, with the speeds arranged in increasing
order.
\section{Main Result}
\begin{theorem}
Let $M = \{m_1,m_2, \ldots, m_n\}$ where $n \geq 2,$ and
$(\frac{m_{j+1}}{m_j})(\frac{n-1}{n+1}) \geq 2$ for each $j = 1,2,
\ldots , n-1$. Then there exists a real number $x$ such that
\[\|m_jx\| \geq \frac{1}{n+1},\]
for each $j = 1,2, \ldots , n$.
\end{theorem}
\begin{proof}
Consider an interval $I = [u, v] = [\frac{1}{m_1(n+1)},
\frac{n}{m_1(n+1)}].$ Clearly, for $x \in I,$ we have $\|m_1x\| \geq
\frac{1}{n+1},$ and $v-u = \frac{1}{m_1}(\frac{n-1}{n+1}) $. Let us
denote the interval $I$ by $I_1.$ We now construct the intervals
$I_2, I_3, \ldots I_n$ satisfying the following properties:
\begin{enumerate}
\item[(a)] $I_1 \supset I_2 \supset I_3 \supset \ldots \supset I_n$
\item[(b)] For $I_j = [u_j, v_j],$ $v_j - u_j = \frac{1}{m_j}(\frac{n-1}{n+1})$
\item[(c)] For each $x \in I_j$ we have $\|m_jx\| \geq \frac{1}{n+1}$
\end{enumerate}
Clearly, $I_1$ satisfies $(b)$ and $(c)$. Inductively, we now define
the $j$th interval $I_j = [u_j, v_j].$ We have
\[m_jv_{j-1} - m_ju_{j-1} = \frac{m_j}{m_{j-1}}\left(\frac{n-1}{n+1}\right) \geq 2.\]
Therefore, there exists an integer $\ell(j)$ such that
\[m_ju_{j-1} \leq \ell(j) < \ell(j) + 1 \leq m_jv_{j-1} \Rightarrow u_{j-1} \leq \frac{\ell(j)}{m_j} < \frac{\ell(j) + 1}{m_j} \leq v_{j-1}.\]
Define,
\[I_j = [u_j, v_j] = \left[\frac{\ell(j) + \frac{1}{n+1}}{m_j}, \frac{\ell(j) + \frac{n}{n+1}}{m_j}\right].\]
It can be seen easily that the interval $I_j$ satisfies all $(a)$,
$(b)$ and $(c).$ Since the intersection of the intervals $I_1, I_2,
\ldots I_n$ is nonempty therefore, we have the theorem.
\end{proof}
In the theorem we have seen that the $n$ runners having their speeds
$r_1,r_2, \ldots, r_n$ with $(\frac{r_{j+1}}{r_j})(\frac{n-1}{n+1})
\geq 2$ satisfy the Lonely Runner Conjecture.
\section{Acknowledgements}
I am very much thankful to the referee for his/her useful
suggestions to present the paper in a better form.
\begin{thebibliography}{5}
\bibitem {serra}
J. Barajas and O. Serra,
\href{http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B75GV-4PCHPN5-2R-1&_cdi=
13104&_user=708524&_orig=browse&_coverDate=08%2F15%2F2007&_sk=999709999&view=c&wchp=dGLbVlb-zSkzk&md5=
5edbf93ab4cd12ec0ddabc7c5592f450&ie=/sdarticle.pdf}
{Regular chromatic number and the lonely runner problem}, {\it Electron. Notes Discrete Math.}, {\bf
29} (2007), 479--483.
\bibitem {barserra}
J. Barajas and O. Serra,
\href{http://www.combinatorics.org/Volume_15/PDF/v15i1r48.pdf}
{The lonely runner with seven runners}, {\it Electron. J. Combin.}, {\bf 15} (2008), Paper \# R48.
\bibitem {betkewills}
U. Betke and J. M. Wills, Untere Schranken f{\"u}r zwei
diophantische Approximations-Funktionen, {\it Monatsh. Math.}, {\bf
76} (1972), 214--217.
\bibitem {bienia}
W. Bienia, L. Goddyn, P. Gvozdjak, A. Seb\H{o} and M. Tarsi, Flows,
view obstructions and the lonely runner, {\it J. Combin. Theory Ser.
B\/}, {\bf 72} (1998), 1--9.
\bibitem {bohman}
T. Bohman, R. Holzman and D. Kleitman,
\href{http://www.combinatorics.org/Volume_8/PDF/v8i2r3.pdf} {Six
lonely runners}, {\it Electron. J. Combin.}, {\bf 8} (2001), Paper \# R3.
\bibitem {cusick}
T. W. Cusick, View-obstruction problems in $n$-dimensional geometry,
{\it J. Combin. Theory Ser. A\/}, {\bf 16} (1974), 1--11.
\bibitem {cusickpom}
T. W. Cusick and C. Pomerance, View-obstruction problems III, {\it
J. Number Theory }, {\bf 19} (1984), 131--139.
\bibitem {goddyn}
L. Goddyn and E. B. Wong,
\href{http://www.westga.edu/~integers/cgi-bin/get.cgi}
{Tight instances of the lonely runner}, {\it Integers\/}, {\bf 6} (2006),
Paper A38.
\bibitem {renault}
J. Renault, View-obstruction: a shorter proof for six lonely
runners, {\it Discrete Math.}, {\bf 287} (2004), 93--101.
\bibitem {wills}
J. M. Wills, Zwei S{\"a}tze uber inhomogene diophantische
Approximation von Irrationalzahlen, {\it Monatsh. Math.\/}, {\bf 71}
(1967), 263--269.
\end{thebibliography}
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\bigskip
\noindent 2000 {\it Mathematics Subject Classification}: Primary
11B50; Secondary 11B75, 11A99.
\noindent \emph{Keywords:} integers, distance from the nearest integer.
\bigskip
\hrule
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\noindent
Received April 16 2009;
revised version received June 4 2009.
Published in {\it Journal of Integer Sequences}, June 5 2009.
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