\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics,amsmath,amssymb} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{epsf} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.5in} \setlength{\textheight}{8.9in} \newcommand{\seqnum}[1]{\href{http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=#1}{\underline{#1}}} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{defin}[theorem]{Definition} \newenvironment{definition}{\begin{defin}\normalfont\quad}{\end{defin}} \newtheorem{examp}[theorem]{Example} \newenvironment{example}{\begin{examp}\normalfont\quad}{\end{examp}} \newtheorem{rema}[theorem]{Remark} \newenvironment{remark}{\begin{rema}\normalfont\quad}{\end{rema}} \newtheorem{note}[theorem]{Note} \newenvironment{Note}{\begin{note}\normalfont\quad}{\end{note}} \vbox {\vspace{6mm}} \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} \end{center} \vskip .2in \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} \bigskip \hrule \bigskip \noindent 2000 {\it Mathematics Subject Classification}: Primary 11B50; Secondary 11B75, 11A99. \noindent \emph{Keywords:} integers, distance from the nearest integer. \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received April 16 2009; revised version received June 4 2009. Published in {\it Journal of Integer Sequences}, June 5 2009. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}