% spellchecked: % last edit: pdh11-10-98 % gallies sent: % gallies corrected: % set in production style: % Section Inroads editor Mauldin % Received 6/8/98 % AMS # 28A78 \documentclass{rae} \usepackage{amsmath,amsthm,amssymb} %\coverauthor{Tam\'as Keleti} %\covertitle{A 1-Dimensional Subset of the Reals that Intersects Each of its Translates in at Most a Single Point} \received{June 8, 1998} \MathReviews{28A78} \keywords{Hausdorff dimension, translation, linearly independent} \firstpagenumber{1} \markboth{Tam\'as Keleti} {A 1-Dimensional Subset of the Reals} \author{Tam\'as Keleti, \thanks{This research was done while the author was visiting the University College London having a Royal Society/NATO Postdoctoral Fellowship award.} Department of Analysis, E\"otv\"os Lor\'and University, M\'uzeum krt. 6-8, Budapest, H-1088, Hungary e-mail: {\tt elek@cs.elte.hu}} \title{A 1-DIMENSIONAL SUBSET OF THE REALS THAT INTERSECTS EACH OF ITS TRANSLATES IN AT MOST A SINGLE POINT} %%%%%% Put Author's Definitions Below Here %%%%%%%%%%%%%%%%%%%%% \def\r{\right} \def\l{\left} \def\tn{{{\left[{{t\over n}}\right]}}} \def\I{[0,1]} \def\d{\partial} \def\i{{{\rm int}}} \def\length{{{\rm length}}} \def\kesz{\ \lower.25mm\vbox{\hrule\hbox{\vrule\kern.75mm {\vbox{\kern1.5mm}}\kern.75mm\vrule}\hrule}} \emergencystretch=0pt \overfullrule=0pt \def\cupkm{\cup_{{k=1}}^m} \def\cupsm{\cup_{{s=1}}^m} \def\sumkm{\sum_{{k=1}}^m} \def\sumsm{\sum_{{s=1}}^m} \def\sumin{\sum_{{i=1}}^n} \def\sumim{\sum_{{i=1}}^m} \def\sumn{\sum_{{n=1}}^\infty} \def\sumj{\sum_{{j=1}}^\infty} \def\sumk{\sum_{{k=1}}^\infty} \def\suml{\sum_{{l=1}}^\infty} \def\cupk{\cup_{{k}}} \def\cupi{\cup_{{i=1}}^\infty} \def\capm{\cap_{{l=1}}^\infty} \def\frac#1#2{{{#1 \over #2}}} \def\perc[#1]{{{#1 \over c}}} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{mainlemma}[theorem]{Main Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{notation}[theorem]{Notation} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newtheorem{question}[theorem]{Question} \newtheorem{facts}[theorem]{Facts} \def\rem{\begin{remark}\rm} \def\erem{\end{remark}} \def\exm{\begin{example}\rm} \def\eexm{\end{example}} \def\dist{{{\rm dist}}} \def\sgn{{{\rm sgn}}} \def\diam{{{\rm diam}}} \def\text#1{\hbox{{\rm #1}}} \def\textsl#1{\hbox{{\sl #1}}} \def\alahuz#1{$\underline{\hbox{{\sl #1}}}$} \def\cl#1{\overline{{#1}}} \def\dst{\displaystyle } \def\ex{\exists} \def\all{\forall} \def\eps{\varepsilon } \def\al{\alpha } \def\be{\beta } \def\de{\delta } \def\ga{\gamma } \def\la{\lambda } \def\si{\sigma} \def\om{\omega } \def\proof{\emph{Proof. }} \def\cont{continuous } \def\fel{{{1 \over 2}}} \def\R{{{\bf R}}} \def\T{{{\bf T}}} \def\Z{{{\bf Z}}} \def\N{{{\bf N}}} \def\Q{{{\bf Q}}} \def\Rn{{{\bf R}^n}} \def\csillag{({\raise -3pt \hbox{*}}) } \def\csillagv{({\raise -3pt \hbox{*}})} \def\ketcsillag{({\raise -3pt \hbox{**}}) } \def\haromcsillag{({\raise -3pt \hbox{***}}) } \def\ketcsillagv{({\raise -3pt \hbox{**}})} \def\haromcsillagv{({\raise -3pt \hbox{***}})} \def\es{\ {\rm and}\ } \def\vagy{\ {\rm or}\ } \def\ees{\quad {\rm and}\quad } \def\vaagy{\quad {\rm or}\quad } \def\su{\subset} \def\sp{\supset} \def\sm{\setminus} \def\de{\delta} \def\L{\Lambda } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \maketitle \begin{abstract} We construct a compact subset of $\R$ with Hausdorff dimension 1 that intersects each of its non-identical translates in at most one point. Moreover, one can make the set to be linearly independent over the rationals. \end{abstract} In 1984 P. Mattila \cite{Ma} constructed compact subsets $A$ and $B$ of $\R$ with Hausdorff dimension 1 such that the intersection of $A$ and any translate of $B$ contains at most one point. In this note we show that - if we allow only non-identical translations - one can also have $A=B$. We call a set of 3 or 4 real numbers \$x_1