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% Section Research editor Mauldin
% Received 10/6/97
% AMS # 28A75 and 28A15
\documentclass{rae}
\usepackage{amsmath,amssymb,amsthm}
%\coverauthor{Marianna Cs\"ornyei}
%\covertitle{On the Affine Sharpness of Heart's Density Theorem}
\received{October 6, 1997}
\author{Marianna Cs\"ornyei\thanks{Research supported by Grants FKFP 0189/1997 and Hungarian
National Foundation for Scientific Research Grant No. T019476.}, E\"otv\"os University, Department of
Analysis, M\'uzeum Krt. 6-8, H-1088 Budapest, Hungary, e-mail:
{\tt csornyei@cs.elte.hu}}
\title{ON THE AFFINE SHARPNESS OF HEART'S DENSITY THEOREM}
\markboth{Marianna Cs\"ornyei}{Affine Sharpness of Heart's Density
Theorem}
\MathReviews{28A75, 28A15}
\firstpagenumber{1}
\newcommand{\pf}{\noindent{\sc Proof.} }
\newtheorem*{theorem}{Theorem}
\newtheorem*{lemma}{Lemma}
{\theoremstyle{definition}\newtheorem*{remark}{Remark}}
%\def\frac#1#2{{{#1}\over{#2}}}
\def\In{I_{n_1n_2\mbox{$\dots$} n_k}}
\def\ln{l_{n_1\mbox{$\dots$} n_k}}
\def\nn{{\bar n}}
\def\nk{{n_1n_2\mbox{$\dots$} n_k}}
\def\ll{{\bar l}}
\def\lr{{l_1l_2\mbox{$\dots$}l_r}}
%\def\qed{\nobreak\hskip3mm\nobreak\vrule width2mm height2mm
%depth0mm}
\def\emp{\emptyset}
\def\nemp{\neq\emp}
%\def\edef{ \mbox{$\buildrel{\rm def} \over =$}}
%\renewcommand{\edef}{\mbox{$\overset{\rm def}{=}$}}
\def\la{\lambda}
\def\eps{\varepsilon}
\def\del{\delta}
\begin{document}
\maketitle
\begin{abstract}
Let $\{I_n\}_{n=1}^\infty$ be a sequence of pairwise disjoint intervals
tending to 0 from the right, such that putting $I_n=(a_n,b_n)$ we have
$\lim\inf {\dfrac{|I_n|}{b_n}}=0$. We prove that
for every Cantor set $C$ there exists some set $X$ of
positive measure, such that for a.e. $x\in X$ there exists a
$c\in C$ for which $(x+cI_n)\cap X=\emptyset$ for infinitely many $n$.
\end{abstract}
V. Aversa and D. Preiss proved in [1] that there exists a sequence of
pairwise disjoint positive intervals $\{I_n\}_{n=1}^\infty$ tending to $0$,
such that putting
$I_n=(a_n,b_n)$ we have
\begin{equation}\label{1}
{\liminf}\frac {|I_n|}{b_n} =0,
\end{equation}
but the differentiation system given by the
translations of this system of intervals has the density
property; that is, for every measurable set $X$ we have
$$\frac{\lambda((x+I_n)\cap X)}{|I_n|} \to 1 \quad ({\rm
a.e.}\, x\in X).$$ In the same paper it is proved that the differentiation
system given by the affine images of this system has the density
property if and only if $(1)$ does not hold. Actually their proof
shows that for every sequence of pairwise disjoint positive
intervals tending to $0$ for which $(1)$ holds, there exists some
set $X$ of positive measure, such that for a.e. $x\in X$ there
exists a $c\in (0,1)$ for which $(x+cI_n)\cap X=\emp$ for
infinitely many $n$.
In this paper, applying entirely different
methods, we prove that the same statement holds even if the
multipliers $c$ are taken only from a given Cantor set, rather
than from $(0,1)$.
\begin{remark}
This implies immediately that if the multipliers $c$ are taken
from a Borel set $B$, then the statement holds if and only
if $B$ is uncountable.
\end{remark}
D. Borwein and S. Z. Ditor proved in [2] that there exists a set
$X$ of positive measure and a sequence $d_n$ of positive numbers
tending to 0 such
that $x+d_n\not\in X$ for infinitely many $n$.
A stronger result is proved in [3]. It is proved that for a
sequence of pairwise disjoint
positive intervals $\{I_n\}_{n=1}^\infty=\{I^1_1,I^1_2,...,I^1_{n_1}$,
$I^2_1,I^2_2,...,I^2_{n_2},...\}$ tending to $0$,
where $I^j_1,...,I^j_{n_j}$ are of the same length and
$\{n_j\}_{j=1}^\infty$ is not bounded,
there exists a Cantor set $C$ of positive
measure such that for a.e. $x\in C$ the interval $x+I_n$ is disjoint
from $C$ for infinitely many $n$. As a generalization of this result
now we prove the following lemma.
\begin{lemma}
Let $\{I_{n_1n_2...n_{k+1}}:\,
n_1=1,\,\, 1\le n_{k+1}\le m_\nk ,\, k=0,1,...\}$ be a system of
open intervals, such that
\begin{enumerate}
\item[(i)] $I_{\nk j}\,\,(1\le j\le m_\nk)$
are pairwise disjoint intervals of the same length;
\item[(ii)] in every sequence $I_{n_1}, I_{n_1n_2},
I_{n_1n_2n_3},...$ the intervals are pairwise
disjoint and with lengths that tend to $0$;
\item[(iii)] For every interval $\In$
we have $\sup_{l_1l_2...}m_{\nk \lr}=\infty.$
\end{enumerate}
Then there exists a Cantor set $C$ of positive measure
such that for a.e. $x\in C$
there exists a sequence $n_1,n_2,...$
such that for infinitely many $k$ the intervals $x+\In$ are
disjoint from $C$.
\end{lemma}
\pf
We construct our Cantor set $C$ by a perfect scheme; that is,
$C=\cap C_n$, where
$C_n$ is the union of non-overlapping closed intervals each ``associated
to" one of the intervals $\In$ of the same length, and
$C_1\supset C_2\supset C_3\supset...$.
For ease of notation we put $l_\nk\overset{\rm def}{=} |\In|$, and
let $k_\nk$ be the least positive integer for which
$(0,k_\nk\cdot l_{\nk j})$ covers
the union of intervals $I_{\nk j}\,\,(1\le j\le m_\nk)$.
Let $C_1=[0,l_1]$, and we say that this interval is associated
to $I_1$. Suppose we have already defined $C_{n-1}$, and let $J$
be one of its intervals, associated to the interval $\In$. We
choose an interval $I_{\nk \lr}$, for which putting $\nn\overset{\rm
def}{=}\nk $,
$\ll\overset{\rm def}{=}\nk \lr $ and $l^*\overset{\rm def}{=} l_{\ll j}$
we have
\begin{equation}\label{2}
\frac{k_{\ll}l^*} {l_\nn} <\frac 1 {n^2}
\end{equation}
and $m_\ll>n^2$. This
implies
$k_\ll>n^2$.
Let $F$ be the set of intervals $((i-
1)l^*,il^*)$ where $i$ is a positive integer and
this interval intersects some of the intervals $I_{\ll j}$. Now,
$F$ has at least $n^2$ elements. We choose $n$ elements, say
with right endpoints $i_1l^*,...,i_nl^*$.
Let $G\overset{\rm def}{=}\{i_1,...,i_n\}$, and
by the greedy algorithm we choose integers $0<
m_1,...,m_q \le k_\ll$ for which the sets $G-m_i$ are pairwise
disjoint (that is, we may choose $m_1=1$, let $m_2$ be the least
integer for which $G-m_1$ and $G-m_2$ are disjoint, etc., etc.)
Each set $G-m_i$ consists of $n$ intervals. Thus $\bigcup_{i=1}^q (G-m_i)$
intersects $G-m$ for at most $qn^2$ different values of $m$.
%thus we have $q\ge \frac{k_\ll}{n^2}$, and
We stop the algorithm
when $\frac{k_\ll}{n^2}\le q\le \frac{k_\ll}{n^2}+1$. (We know
$k_\ll> n^2$).
Let $$A\overset{\rm def}{=}\bigcup_{i=m_p-2,m_p-
1,m_p,m_p+1,\,\,1\le p\le q}((i-1)l^*,il^*),$$
and let
$$B\overset{\rm def}{=} \bigcup_{i=m_p-i_t, 1\le p\le q, 1\le t\le
n}[(i-1)l^*,il^*]\setminus A.$$ Now, for every $((i-1)l^*,il^*)\subset B$
there exists an $i_t$ such that
$$x+((i_t-2)l^*,(i_t+1)l^*)\subset A;$$
that is, there is a $j$ such that $x+I_{\ll j}\subset A$.
The measure of $A$ is ``small", and the measure of $B$ is ``large":
\begin{equation}\label{3}
\frac{k_\ll l^*}{n^2}\le ql^*\le \lambda(A)\le
4ql^*\le 8\frac{k_\ll l^*}{n^2},
\end{equation}
\begin{equation}\label{4}
\lambda(B)\ge nql^*-\lambda(A)\ge \frac{k_\ll
l^*}{n}- 8\frac{k_\ll l^*}{n^2},
\end{equation}
and
$A\subset (-2l^*,(k_\ll+1)l^*)\subset (-2k_\ll l^*, 2k_\ll l^*)$,
$B\subset [-k_\ll l^*, k_\ll l^*]\subset (-2k_\ll l^*, 2k_\ll
l^*)$.
Now we construct $C_n$. For the interval $J=[a,b]$ associated to
$\In$ we consider the subintervals $[a+(4s-4)k_\ll l^*,a+4sk_\ll
l^*]\,\,(s=1,2,...)$, and we delete the remainder of the interval
$(a+4sk_\ll l^*,b)$ of length less than $4k_\ll l^*$, and we
delete the sets $a+(4s-2)k_\ll l^*+A\subset a+[(4s-4)k_\ll l^*,
a+4sk_\ll l^*]$, and possible isolated points. For every interval
$I=a+(4s-2)k_\ll l^*+((i-1)l^*,il^*)$, where $((i-
1)l^*,il^*)\subset B$, there is an interval $I_{\ll j}$ for which
$x+I_{\ll j}$ lies in the deleted set $x+(4s-2)k_\ll l^*+A$. The
interval $a+(4s-2)k_\ll l^*+[(i-1)l^*,il^*]$ is said to be
associated to one of these intervals $I_{\ll j}$, and the other
non-deleted intervals of form $a+(4s-2)k_\ll l^*+[(i-1)l^*,
il^*]$ are said to be associated to an interval $I_{\ll j}$ where
$j$ is chosen arbitrarily. This process yields a Cantor set $C$.
According to $\eqref{2}$ and $\eqref{3}$ the set $C$ constructed is of
positive measure, and according to $\eqref{4}$ the set of points
of $C$ contained in a set of the form $a+(4s-2)k_\ll l^*+B$
only finitely often is of measure $0$. \qed
\begin{theorem}
For a sequence of pairwise disjoint intervals
$\{J_n\}_{n=1}^\infty$ and a Cantor set $E$
condition (1) implies that there exists a Cantor set $C$ of
positive measure, such that for a.e. $x\in C$ there exists a
$c\in E$ for which $x+cJ_n$ is disjoint from $C$ for
infinitely many $n$.
\end{theorem}
\pf
Let $J_n=(a_n,b_n)$ satisfy (1); that is, $\lim\inf\frac{b_n-
a_n}{b_n}=0$. Let $E=\cap E_n$ be a Cantor set,
where $E_1\supset E_2\supset...$ are the
closed sets of a perfect scheme defining $E$, each is the union
of finitely many closed intervals.
We construct a system of intervals $\In$ satisfying the
conditions of the lemma, together with another system of closed
intervals $L_\nk\subset E_k$ for which $L_\nk \supset L_{\nk
n_{k+1}}\supset...,\,\,{\rm int} L_\nk\cap E\nemp$, and for every
$\nk$ there exists an interval $J_n$ of our sequence, such that
for every $c\in L_\nk$ we have $cJ_n\subset I_\nk$.
Having these systems of intervals defined the
theorem easily follows because by the Lemma
there exists a Cantor set $C$ of positive
measure such that for a.e. $x\in C$ there exist a branch
$I_{n_1}, I_{n_1n_2}...$ and infinitely many $k$
for which $C\cap (x+\In)=\emp$. Hence for $c\in\bigcap_k
L_\nk\subset\bigcap_k E_k=E$ we have $C\cap (x+cJ_n)=\emp$ for
infinitely many $n$.
Let $L_1=[c_1,d_1]\subset E_1$ such that $(c_1,d_1)\cap E\nemp$,
and put $I_1\overset{\rm def}{=}(c_1a_1,d_1b_1)$.
Suppose we have already defined the intervals $\In$ and
$L_\nk=[c,d]$, where $(c,d)\cap E\nemp$.
We choose $\del>0$ and also
points $\la^1,\la^2,...,\la^k\in(c,d)\cap E$ such that
$\la^1<\la^2<...<\la^k$, $(\la^j-\del,\la^j)\subset E_k$, and for
each $\la^j$ and for every $\eps>0$ we have $(\la^j-
\eps,\la^j)\cap E\nemp$. We can choose $J_n=(a_n,b_n)=(a,b)$ such
that $(0,db)\cap\In=\emp$, and putting $$b^j\overset{\rm def}{=}
\la^jb,\,\,a^j\overset{\rm def}{=}\la^jb-d(b-a)$$ the intervals
$[a^1,b^1], [a^2,b^2],...,[a^k,b^k]$ are pairwise disjoint subintervals of
$(ca,db)$ of length $d(b-a)$, and $\frac{a^j}a>\la^j-
\del$. Indeed, we observe that the two conditions
$ca\la^j-\del$ hold if and only if
$\frac{|J_n|}{b_n}<\min(\frac{\la^1-c}{d-c},
\frac{\la^{j+1}-\la^j} d,\frac{d-\la^j+\del} \del)$.
Let
$$I_{\nk j}\overset{\rm def}{=}
[a^j,b^j]\,\,(j=1,2,...,m_\nk\overset{\rm def}{=} k)$$
\noindent and %let $
$$L_{\nk j}\overset{\rm def}{=} [\frac{a^j}a,\frac{b^j}b].$$
We need
$$c<\frac{a^1}a<\frac{b^1}b<\frac{a^2}a<...<\frac{b^k}b