Content-Type: multipart/mixed; boundary="-------------0009270459836" This is a multi-part message in MIME format. ---------------0009270459836 Content-Type: text/plain; name="00-378.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-378.keywords" harmonic measure, simple random walk, multifractal spectrum ---------------0009270459836 Content-Type: application/x-tex; name="hdff4.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="hdff4.tex" \documentclass{article} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{latexsym} % %\magnification=magstep1 \parindent=1em \baselineskip 15pt \textwidth=12.3cm \textheight=18.5cm \newcommand{\qed}{\hspace*{\fill} $\Box$} \def\endrem{} \def\colon{{:}\;} \newcommand{\be}[1]{\begin{equation} \label{#1}} \newcommand{\dpartial[2]}{\frac{\partial {#1}}{\partial {#2}}} \newcommand{\dd[2]}{\frac{d {#1}}{d {#2}}} \newcommand{\dps}{\displaystyle} \def\gdw{\Leftrightarrow} \def\gdwl{\Longleftrightarrow} \def\impl{\Rightarrow} \def\impll{\Longrightarrow} \def\lto{\longrightarrow} \def\lmap{\longmapsto} \def\ol{\overline} \def\ul{\underline} \newcommand{\wt}[1]{\widetilde{#1}} \newcommand{\norm}[1]{\| {#1}\| } \def\strich{$^\prime$} \def\charfct{1\hspace{-2pt} {\rm l}} \DeclareMathOperator{\divv}{div} \DeclareMathOperator{\supp}{supp} \def\schnitt{\cap} \def\Schnitt{\bigcap} \def\verein{\cup} \def\Verein{\bigcup} \def\oder{\vee} \def\und{\wedge} \def\fa{\forall} \def\ex{\exists} \def\a{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon} \def\l{\lambda} \def\s{\sigma} \def\vphi{\varphi} \def\vtheta{\vartheta} \def\cC{{\cal C}} \def\cT{{\cal T}} \def\K{{\mathbb K}} \def\R{{\mathbb R}} \def\Q{{\mathbb Q}} \def\N{{\mathbb N}} \def\Z{{\mathbb Z}} \def\C{{\mathbb C}} \def\P{{\mathbb P}} \def\E{{\mathbb E}} \newtheorem{Theorem}{Theorem} \newtheorem{Lemma}[Theorem]{Lemma} \begin{document} % \title{Quantitative estimates of discrete harmonic measures} % \author{E. Bolthausen and K. M\"{u}nch-Berndl\footnote{Supported by Swiss NF grant 20-55648.98}\\ {\normalsize Institut f\"{u}r Mathematik, Universit\"{a}t Z\"{u}rich,}\\ {\normalsize Winterthurer Str.\ 190, 8057 Z\"{u}rich, Switzerland}} % \date{May 2, 2000} % \maketitle % \begin{abstract} A theorem of Bourgain states that the harmonic measure for a domain in $\R^d$ is supported on a set of Hausdorff dimension strictly less than $d$ \cite{Bourgain}. We apply Bourgain's method to the discrete case, i.e., to the distribution of the first entrance point of a random walk into a subset of $\Z ^d$, $d\geq 2$. By refining the argument, we prove that for all $\b>0$ there exists $\rho (d,\b )N(d,\b )$, any $x \in \Z^d$, and any $A\subset \{ 1,\dots , n\}^d$ $$ | \{ y\in\Z^d\colon \nu_{A,x} (y) \geq n^{-\b} \}| \ \leq \ n^{\rho(d,\b )} , $$ where $\nu_{A,x} (y)$ denotes the probability that $y$ is the first entrance point of the simple random walk starting at $x$ into $A$. Furthermore, $\rho$ must converge to $d$ as $\b \to \infty$. \end{abstract} % \section{Introduction} Let $(S_n)_{n\in \N}$ be a simple random walk in $\Z^d$ starting at $x\in \Z^d$, i.e., $S_0=x$ and $$ \P^x(S_{n+1} - S_n = e) = \frac 1{2d}, \quad \norm{e}=1, \quad n\in\N. $$ ($\norm{ \ . \ }$ denotes the Euclidian distance, i.e., $\norm{x} = \sqrt{x_1^2+\dots +x_d^2}$.) For $A\subset \Z^d$, $A\neq \emptyset$, we denote by $\tau_A$ the time of the first entrance of $S$ to $A$: $$ \tau_A=\inf \{n\geq 0\colon S_n\in A\} . $$ The harmonic measure for $A$ of a set $B\subset \Z^d$ evaluated at $x\in\Z^d$ is defined as $$ \omega (A,B,x) = \P^x (\tau_A<\infty ,\ S_{\tau_A} \in B). $$ Clearly, for $x\in A$, $\omega (A,B,x) = \charfct_B(x)$. For fixed $A\subset \Z^d$ and $x\in \Z^d$, $\omega (A,\ .\ ,x)$ is a measure on $\Z^d$ with total mass $\omega(A,\Z^d,x)= \omega(A,A,x) = \P^x (\tau_A<\infty )\in (0, 1]$. We denote by $\nu_{A,x}(y)=\omega (A,\{ y\}, x)$ its density. For $x\in A^c = \Z^d\setminus A$, $\omega (A,B,\ .\ )$ is a harmonic function, $$ \Delta \omega (A,B,x)= \frac 1{2d} \sum_{\norm{e}=1}\omega (A,B,x+e) - \omega (A,B,x) = 0. $$ We shall prove the following theorem: \medskip \noindent {\bf Theorem.} {\it (A) For all $\b>0$ there exists $\rho (d,\b )N(d,\b )$, any $x \in \Z^d$, and any $A\subset Q^d(n) = \{ 1,\dots , n\}^d$ $$ | \{ y\in\Z^d\colon \nu_{A,x} (y) \geq n^{-\b} \}| \ \leq \ n^{\rho(d,\b )} . $$ (B) For all $\rho \ n_K^{\rho} . $$ } \medskip (For $A\subset \Z^d$, $|A|$ denotes the number of points of $A$.) \medskip \noindent {\bf Remarks.} (1) If $x \in A$, the statement of Theorem (A) is trivial. Therefore we only consider $x \in A^c$. The proof of Theorem (A) is to a large extent an adaptation of Bourgain's proof \cite{Bourgain} that the harmonic measure for a domain in $\R^d$ is supported on a set of Hausdorff dimension strictly less than $d$ to the discrete case, and the proof of Theorem (B) is inspired by Jones and Makarov \cite{JM} who also treat continuous harmonic measure. \endrem (2) The analogous theorems hold for harmonic measure conditioned on the event that $A$ is reached, and also for harmonic measure from infinity: Let $$ \bar\nu_{A,x} (y) = \P^x (S_{\tau_A} =y|\; \tau_A<\infty ) , $$ and $$ \bar\nu_{A,\infty} (y) = \lim_{\| x\| \to \infty}\bar\nu_{A,x} (y). $$ (See for example \cite{Lawlerbook}, Chapter 2.1 for the existence of $\bar\nu_{A,\infty}$.) Then we have {\it (A') For all $\b>0$ there exists $\rho (d,\b )N(d,\b )$, any $x \in \Z^d$, and any $A\subset Q^d(n) = \{ 1,\dots , n\}^d$ $$ | \{ y\in\Z^d\colon \bar\nu_{A,x} (y) \geq n^{-\b} \}| \ \leq \ n^{\rho(d,\b )} , $$ } and {\it (A'') For all $\b>0$ there exists $\rho (d,\b )N(d,\b )$ and any $A\subset Q^d(n) = \{ 1,\dots , n\}^d$ $$ | \{ y\in\Z^d\colon \bar\nu_{A,\infty } (y) \geq n^{-\b} \}| \ \leq \ n^{\rho(d,\b )} . $$ } For (A'), note first that for $d=2$, $\P^x (\tau_A<\infty ) =1$ for all $x$ and $A$ by recurrence and therefore $\bar\nu_{A,x} = \nu_{A,x}$. For $d\geq 3$, we have a lower bound on the hitting probability $\P^x (\tau_A<\infty ) $ for $x$ in a neighborhood of $Q^d(n)$, \begin{equation} \P^x (\tau_A<\infty ) \geq \P^x (\tau_{\{ z\} }<\infty ) = \frac {G(x-z)}{G(0)} \geq \frac {c_2}{G(0)} \| x-z\| ^{2-d} \geq c(a,d) n^{2-d}\label{eq1} \end{equation} for all $z\in A$ and $x\in U^d(an) = \{ -an, \dots, (a+1)n\} ^d$, where $G$ is the Green's function which satisfies (\ref{G}), see Section \ref{Sect:2} below. For more distant $x$, $\bar \nu_{A,x}$ doesn't change a lot any more: For $d\geq 2$, there exist constants $C_1(d)$ and $C_2(d)$ such that for all $A\subset Q^d(n)$, $y\in A$, $x\in (U^d(an))^c$ with $a\geq 2\sqrt d$ \begin{equation} C_1 \bar \nu_{A,x} (y) \leq \bar \nu_{A,\infty} (y) \leq C_2 \bar \nu_{A,x} (y), \label{eq2} \end{equation} see \cite{Lawlerbook}, Chapter 2.1. From (\ref{eq1}) and (\ref{eq2}), (A') follows, and (A'') follows from (A') with (\ref{eq2}). Similarly we have the analogs of Theorem (B). \endrem (3) Our theorem improves a result of Benjamini \cite{Benjamini}. In fact, it implies the following weaker statement (which is still stronger than \cite{Benjamini}): There exists $\rho (d)0$ there is an $N(\e)$ such that for any $n>N(\e)$, any $x \in \Z^d$, and any $A\subset Q^d(n) = \{ 1,\dots , n\}^d$ there is a set $\tilde A\subset A$ with $$ \omega (A, \tilde A, x) > \omega(A,A,x)-\e \quad \text{and} \quad |\tilde A| <\e n^\rho . $$ The analogous statements hold for harmonic measure conditioned on the event that $A$ is reached, and also for harmonic measure from infinity. Note that it is in general impossible that $\tilde A$ carries the full mass: Considering for example (for even $n$) $A= \{ 1,3,5,\dots , n-1\}^d$, the only set having full mass (for $x\not\in A$) is $A$, and $|A|= (n/2)^d$. \endrem (4) The dependence of the exponent $\rho$ on $\b$ for 2-dimensional simple random walk paths $A$ (the ``multifractal spectrum of the harmonic measure for $A$'') has been studied by Lawler \cite{Lawlerpreprint}. Also for $d=2$, there is another result of Lawler \cite{Lawler} which gives more information on the support of harmonic measure from infinity $\bar\nu_{A,\infty}$ for connected sets. \endrem \section{Proof} \subsection{Proof of Theorem (B)} Take $n_K = 2^K$. Delete from $\{ 1,2,\dots ,n_K\}$ the central $\d 2^K$ points, from the remaining two intervals of length $(1-\d) 2^{K-1}$ the central $\d (1-\d) 2^{K-1}$ points, and so on, $k$ $(2^{K\rho}$ and $\nu_{A_K,x_K} (y) >2^{-K\b}$. This is achieved for large enough $K$ by putting $\d$ such that $\rho = d + 3(d-1)\log (1-\d) / \log 2$, $\b$ such that $\b-d+1$ $=$ $4 \log c/(\d \log 2)$, and $k=\g K$ with $\g = \log \left[ 2(1-\d)^{3(d-1)}\right] $ $/\log \left[ 2(1-\d)^{d-1}\right]$. \subsection{Discrete Hausdorff measure} For bounded sets $A\subset \Z^d$, consider coverings of $A$ by a countable number of balls $B_\a$ in $\Z^d$ with center $z_\a$ and radius $r_\a$, $A\subset \Verein_\a B_\a$ with $$ B_\a = \{ x\in \Z^d\colon \|x-z_\a \| \leq r_\a \} . $$ For $0<\rho \leq d$ we define $$ h_\rho (A) = \inf \left\{ \sum_\alpha |B_\alpha |^{\rho / d} ; B_\alpha \text{ ball}, A\subset \Verein_\a B_\a \right\} . $$ Furthermore, consider a net of $l$-adic cubes: $\cC_0= \Z^d$, $\cC_1 = \{ $cubes $C\subset \Z^d$ with side length $|C|^{1/d} = l$ and lower corner $c=(k_1l, k_2l, \dots , k_dl)$ with $k_i\in \Z \}$, $$ \cC_j = \{ C\subset \Z^d\colon C=\{z\in\Z^d\colon k_il^j\leq z_i<( k_i+1)l^j ,k_i\in\Z, i=1\dots d\}\}, $$ and $\cC = \Verein_{j\in\N} \cC_j$. Analogously to $h_\rho$ we define $$ m_\rho (A) = \inf \left\{ \sum_\alpha |C_\alpha |^{\rho / d} ; C_\alpha \in\cC , A\subset \Verein_\a C_\a \right\} . $$ Clearly, there exist two positive constants $t_1(d)$ and $t_2(d,l,\rho)$ such that for all $A\subset \Z^d$ \begin{equation}\label{hrhomrho} h_\rho (A) \leq t_1 (d) m_\rho (A) \end{equation} and \begin{equation}\label{mrhohrho} m_\rho (A)\leq t_2(d,l, \rho) h_\rho (A) . \end{equation} By considering for example a ball of radius $\sqrt l$, one sees that the dependence of $t_2$ on $l$ cannot be removed. A possible choice is \begin{equation}\label{t2} t_2= 8^d l^{d-\rho}. \end{equation} Analogously to Theorem 1 in Carleson \cite{Carleson}, p.7, (see also \cite{Tsuji}, Chapter III.4) we have the following Lemma: \begin{Lemma}\label{L:Carleson} There are constants $t_3$ and $t_4$, depending only on $d$, such that for every bounded set $A\subset \Z^d$ there is a discrete measure $\mu$ supported on $A$ with \begin{equation}\label{ballest} \mu(B) \leq t_3|B|^{\rho /d} \quad \text{for all balls }B\subset \Z^d \end{equation} and \begin{equation}\label{massest} \mu(A) \geq t_4 \, h_\rho(A). \end{equation} \end{Lemma} \noindent {\bf Proof.} Start the construction of $\mu$ by putting $\mu_0(\{ x\} )=1$ for all $x\in A$ and $\mu_0(\{ x\})=0$ for $x\in A^c$. Choose your favorite $l$ and consider the cubes of $\cC_1$. If for some $C\in \cC_1$ $\mu_0(C)>|C|^{\rho/d}$, reduce the density on the points of $C$ uniformly such that $\mu_1(C)=|C|^{\rho/d}$. Continue in this way. After finitely many steps no further reduction will occur, since $\mu_k(C)\leq |A|$ for all $C$ and $k$ and $|A| < l^{K\rho}$ for $K$ large enough. Put $\mu=\mu_K$. $\mu$ satisfies $$ \mu (C)\leq |C|^{\rho/d} \quad \text{for all } C\in \cC $$ and therefore we have (\ref{ballest}). From the construction of $\mu$, each point $a\in A$ is contained in a cube $C_\a$ with $\mu(C_\a) = |C_\a|^{\rho/d}$. If there are several such cubes, choose the largest one. With this (disjoint) covering $\{ C_\a\}$ we obtain $$ \mu (A) = \sum_\a \mu (C_\a) = \sum_\a |C_\a|^{\rho/d}\geq m_\rho(A) \geq \frac 1{t_1(d)}\, h_\rho(A) $$ with (\ref{hrhomrho}). This proves (\ref{massest}).\qed $\mu$ puts more mass on boundary points than on interior points. Thus it is useful for estimating the harmonic measure, which is concentrated on the boundary. \subsection{Estimate of the trapping probability}\label{Sect:2} Another useful quantity to estimate the harmonic measure in $d\geq 3$ is the Green's function $G$, $G(x)$ being the expected number of visits to $x$ of the random walk starting at 0, $$ G(x) = \E ^0\left(\sum_{j=0}^\infty \charfct_{\{x\}}(S_j)\right) = \sum_{j=0}^\infty\P^0 (S_j=x). $$ $G$ is harmonic in $\Z^d\setminus \{ 0\}$, $\Delta G(x)= -\delta (x)$, and $G$ has the following asymptotic behavior: $$ \lim_{\norm{x}\to \infty} \frac {G(x)}{a_d \norm{x}^{2-d}} =1, $$ where $a_d= 2/((d-2)\omega _d) $, and $\omega_d$ is the volume of the unit ball in $\R^d$ (see for example \cite{Lawlerbook}, p.31). This implies that there are constants $c_1$ and $c_2$ ($00$ such that for all $a\in Q_*$ \begin{equation} \omega (A\verein Q^c, A\schnitt Q, a) \geq \tilde c \, \frac {h_\rho(A\schnitt Q_*)}{|Q_*|^{\rho/d}} \label{omegaest}\end{equation} \end{Lemma} \noindent {\bf Proof.} If $A\schnitt Q_* = \emptyset$, (\ref{omegaest}) holds trivially. Let now $A\schnitt Q_* \neq \emptyset$ and let $\mu$ be the measure on $A\schnitt Q_*$ from Lemma \ref{L:Carleson}. We treat first the case $d\geq 3$. Consider the function $u\colon \Z^d \to \R^+$, $$ u (x) = \sum_{y\in A\schnitt Q_*} G(x-y)\, \mu(\{ y\}). $$ $u$ is harmonic in $(A\schnitt Q_*)^c$. For $x\in Q_*$ and $y\in Q_*$, $\norm{x-y}\leq |Q_*|^{1/d}\sqrt d$, and therefore with (\ref{G}) \begin{equation}\label{u*} u(x)\geq c_2 d ^{(2-d)/2} |Q_*|^{(2-d)/d} \mu(A\schnitt Q_*) \quad \text{for } x\in Q_* . \end{equation} For $x\in Q^c$ and $y\in Q_*$, $$ \norm{x-y}\geq \frac{|Q|^{1/d} - |Q_*|^{1/d}}2 \geq \frac {1-q}{2q} |Q_*|^{1/d} $$ and therefore with (\ref{G}) \begin{equation}\label{ud} u(x)\leq c_1 \left( \frac {1-q}{2q}\right)^{2-d} |Q_*|^{(2-d)/d} \mu(A\schnitt Q_*) \quad \text{for } x\in Q^c . \end{equation} Furthermore, for all $x\in \Z^d$ \begin{equation}\label{us} u(x)\leq c_3 |Q_*|^{(2+\rho -d)/d} , \end{equation} where $c_3$ depends only on $d$. This is seen as follows: First of all, with (\ref{G}), $$ \sup_{x\in\Z^d} u(x) = \sup_{x\in B(Q_*)} u(x) , $$ where $B(Q_*)$ is a ball with the same center as $Q_*$ and radius $a/2 \sqrt d |Q_*|^{1/d}$ with suitably chosen $a$ ($a = 1+2(c_1/c_2)^{1/(d-2)}$). Now, for $x\in B(Q_*)$, $$ u(x) = \sum_{k=1}^{a \sqrt d |Q_*|^{1/d}} \sum _{y\in \tilde B_k(x) }G(x-y)\, \mu(\{ y\}) , $$ where $\tilde B_k(x) = \{ y\in \Z^d\colon k-1\leq \| x-y\| 0$ be small enough. Then for all $l$ there exists $\rho 0$ and $n>N(\b)$ (to be chosen below). Let $A\subset Q^d(n)$, $x\in\Z^d$, and let $k^*\in \N$ be such that $l^{k^*}\geq n>l^{k^*-1}$. To the lower bound $N(\b)$ there will correspond a $K^*$ such that $N(\b)=l^{K^*}$. We construct Bourgain's tree $\cT$: starting with $C_0=\{ 1, \dots , l^{k^*}\}^d\in \cC_{k^*}$, we associate to each (L)-cube $C\in \cC_j$ its $l^d$ subcubes in $\cC_{j-1}$, and to each (H)-cube we associate a family $\{ C_\a\}$ with $C_\a \subset C$, $A\schnitt C\subset \Verein _\a C_\a$, and $\sum_\a |C_\a|^{\rho /d}<|C|^{\rho /d}$ (which exists according to Lemma \ref{L:B2}). The elements of the tree are labeled by complexes $\g = (\g_1, \dots , \g_k)$: $C_0$ has the label $\g = (\g_1) = (0)$, its descendants have the label $\g = (\g_1, \g_2) = (0, \g_2)$, and so on. We stop the decomposition when the cube is in $\cC_1$ or $\cC_0$ (because then Lemma \ref{L:B2} doesn't apply any more). Thus each branch is at most $k^*$ long. Denote by $\g |k$ the restriction of $\g$ to the first $k$ digits. If $\tilde k$ is the length of $\g$, we call $C_{\g |1},C_{\g |2}, \dots, C_{\g |\tilde k-1}$ the ``ancestors'' of $C_\g$. Let $\cT^*$ denote the set of the labels of the final cubes. We have \begin{equation}\label{Aueberdeck} A\subset \Verein_{\g\in \cT^*}C_\g . \end{equation} Given a maximal element $\g \in\cT^*$ of length $\tilde k$, we denote by $\tau_k$ the length of the label of the $k$-th (L)-cube appearing in the sequence $C_{\g |1},C_{\g |2}, \dots$ of ancestors of $C_\g$, i.e., $C_{\g |\tau_k}$ is the $k$-th (L)-cube, and $\tau_1<\tau_2<\dots < \tilde k$. ($\tau_k =\infty $ and $\g | \tau_k = \g$ if there are less than $k$ (L)-cubes in the sequence $C_{\g |1},C_{\g |2}, \dots$ of ancestors of $C_\g$.) {\bf (a) Inner cubes.} The subcubes $C_{\g |\tau_k+1}$ of an (L)-cube $C_{\g |\tau_k}$ are distinguished according to whether they lie in $(C_{\g |\tau_k})_{\bar l}$ or not. If $x\in (A\verein C)^c$, the number of subcubes which lie in $(C_{\g |\tau_k})_{\bar l}$ is $(l-2\bar l)^d= (2/3)^d l^d$, and if $x\in C\setminus A$, the number of subcubes which lie in $(C_{\g |\tau_k})_{\bar l}$ is simply estimated as $\geq (l-2\bar l)^d - (2\bar l+1)^d \geq p l^d$ with $p= (2/3)^d - (1/2)^d$. To have a fixed proportion of ``inner'' subcubes (this simplifies the argument in part (c) below), we shall choose for any (L)-cube $pl^d$ subcubes from those subcubes $C_{\g |\tau_k+1} \subset (C_{\g |\tau_k})_{\bar l}$ to call them ``inner'' subcubes. Let $ k_1 = k^*/3$ and $k_2= (p/2) k_1$. Let \begin{eqnarray*} \cT_< & = & \{ \g\in \cT^* \colon \tau_{ k_1} (\g) = \infty \},\\ \cT_i & = & \{ \g\in \cT^* \colon \tau_{ k_1} (\g) < \infty ,\\ & & \text{ at least }k_2 \text{ of } C_{\g |\tau_1+1},C_{\g |\tau_2+1}, \dots,C_{\g |\tau_{k_1}+1} \text{ are inner}\}, \end{eqnarray*} and $\cT_{o} = \cT^* \setminus (\cT_< \verein \cT_i)$. If $C_{\g |\tau_k+1}$ is inner, we have from Lemma \ref{L:B2} $$ \omega (C_{\g |\tau_k+1}) \leq \omega \left( (C_{\g |\tau_k})_{\bar l}\right) \leq \frac {(1-c_4\d)^{\bar l-1}}{c_4\d}\ \omega(C_{\g |\tau_k}), $$ and if not, then in any case $$ \omega (C_{\g |\tau_k+1}) \leq \omega(C_{\g |\tau_k}). $$ Then for $y\in \Verein_{\g\in \cT_i}C_\g$ we have (with $\g$ such that $y\in C_\g$) \begin{eqnarray*} \nu_{A,x} (y)& \leq & \omega(C_\g) \leq \omega (C_{\g | \tau_{k_1}+1})\leq \left( \frac {(1-c_4\d)^{\bar l-1}}{c_4\d}\right) ^{k_2} \omega(C_{\g |\tau_1})\\ & \leq & \left( \frac {(1-c_4\d)^{\bar l-1}}{c_4\d}\right) ^{k_2}. \end{eqnarray*} Now choose $l$ such that $$ \left( \frac {(1-c_4\d)^{\bar l-1}}{c_4\d}\right) ^{k_2} < l^{-k^*\b } , $$ i.e., $$ \frac p6 \left( \frac l6 -1\right)\log \frac 1{1-c_4\d} -\frac p6 \log\frac 1{c_4\d} > \b \log l. $$ Then $$ \Verein_{\g\in \cT_i}C_\g \subset \{ y\in \Z^d\colon \nu_{A,x} (y) < n^{-\b} \}. $$ With (\ref{Aueberdeck}) we obtain $$ \{ y\in\Z^d\colon \nu_{A,x} (y) \geq n^{-\b} \} \subset \Verein_{\g\in \cT_<\verein \cT_o}C_\g . $$ We shall show that $\sum_{\g \in \cT_<} |C_\g| \leq n^{-\tilde \rho}/2$ and $\sum_{\g \in \cT_o} |C_\g| \leq n^{-\tilde \rho}/2$ with $\tilde \rho = (\rho + d)/2$, where $\rho 8 1 false[8 0 0 1 0 0]{}imagemask}{/$bkg true def}ifelse}if}ifelse @gr $wid 0 gt $hei 0 gt and{$pn cvlit load aload pop /$pd xd 3 -1 roll sub/$phei xd exch sub/$pwid xd $wid $pwid div ceiling 1 add /$tlx xd $hei $phei div ceiling 1 add/$tly xd $psx 0 eq{@tv}{@th}ifelse}if @gr @np/$bkg false def}bd/@dlt{$fse $fss sub/nff xd $frb dup 1 eq exch 2 eq or{ $frt dup $frc $frm $fry $frk @tc 4 copy cmyk2rgb rgb2hsb 3 copy/myb xd/mys xd /myh xd $tot $toc $tom $toy $tok @tc cmyk2rgb rgb2hsb 3 1 roll 4 1 roll 5 1 roll sub neg nff div/kdb xd sub neg nff div/kds xd sub neg dup 0 eq{pop $frb 2 eq{.99}{-.99}ifelse}if dup $frb 2 eq exch 0 lt and{1 add}if dup $frb 1 eq exch 0 gt and{1 sub}if nff div/kdh xd}{$frt dup $frc $frm $fry $frk 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1.00 K 0 4.824 4.824 0.000 @w 2336.69 2331.79 m 2393.42 2331.79 L S @rax 2419.34 2189.66 2436.91 2206.94 @E [0.07200 0.00000 0.00000 0.07200 2419.34390 2190.02391] @tm 0 O 0 @g 0.00 0.00 0.00 1.00 k e /_R3-Times-Italic 347.00 z 0 0 (C) @t T @rax 2418.34 2326.46 2436.34 2348.71 @E [0.07200 0.00000 0.00000 0.07200 2418.33590 2331.79190] @tm 0 O 0 @g 0.00 0.00 0.00 1.00 k e /_R3-Times-Italic 347.00 z 0 0 (Q) @t T @rax 2414.38 2341.22 2441.59 2373.84 @E [0.07200 0.00000 0.00000 0.07200 2421.21590 2348.78390] @tm 0 O 0 @g 0.00 0.00 0.00 1.00 k e /_R3-Times-Italic 347.00 z 0 0 (\176) @t T @rax 2419.34 2218.03 2436.91 2235.31 @E [0.07200 0.00000 0.00000 0.07200 2419.34390 2218.39190] @tm 0 O 0 @g 0.00 0.00 0.00 1.00 k e /_R3-Times-Italic 347.00 z 0 0 (C) @t T @rax 2435.90 2215.22 2442.67 2227.39 @E [0.07200 0.00000 0.00000 0.07200 2435.90390 2215.22390] @tm 0 O 0 @g 0.00 0.00 0.00 1.00 k e /_R3-Times-Roman 250.00 z 0 0 (1) @t T @rax 2368.08 2359.87 2390.40 2392.49 @E [0.07200 0.00000 0.00000 0.07200 2373.69590 2367.43190] @tm 0 O 0 @g 0.00 0.00 0.00 1.00 k e /_R3-Times-Italic 347.00 z 0 0 (y) @t T @rax %Note: Object 2362.54 2364.62 2367.65 2369.81 @E 0 O 0 @g 0.00 0.00 0.00 1.00 k 1 J 1 j [] 0 d 0 R 0 @G 0.00 0.00 0.00 1.00 K 0 3.384 3.384 0.000 @w 2365.13 2369.81 m 2366.50 2369.81 2367.65 2368.66 2367.65 2367.22 c 2367.65 2365.78 2366.50 2364.62 2365.13 2364.62 c 2363.69 2364.62 2362.54 2365.78 2362.54 2367.22 c 2362.54 2368.66 2363.69 2369.81 2365.13 2369.81 c @c B @rax %Note: Object 2395.37 2199.89 2413.94 2200.10 @E 1 J 1 j [] 0 d 0 R 0 @G 0.00 0.00 0.00 1.00 K 0 1.656 1.656 0.000 @w 2413.94 2199.89 m 2397.67 2200.10 L S @j 0.00 0.00 0.00 1.00 K 0.00 0.00 0.00 1.00 k 0 @g 0 @G [] 0 d 0 J 0 j 0 R 0 O 0 1.73 1.73 0 @w 2403.14 2195.57 m 2396.38 2200.10 L 2403.29 2204.42 L S @J @rax %Note: Object 2396.23 2340.43 2413.01 2340.72 @E 1 J 1 j [] 0 d 0 R 0 @G 0.00 0.00 0.00 1.00 K 0 1.656 1.656 0.000 @w 2413.01 2340.58 m 2398.54 2340.58 L S @j 0.00 0.00 0.00 1.00 K 0.00 0.00 0.00 1.00 k 0 @g 0 @G [] 0 d 0 J 0 j 0 R 0 O 0 1.73 1.73 0 @w 2404.08 2336.18 m 2397.24 2340.58 L 2404.08 2344.97 L S @J @rax %Note: Object 2367.94 2226.89 2413.73 2227.10 @E 1 J 1 j [] 0 d 0 R 0 @G 0.00 0.00 0.00 1.00 K 0 1.656 1.656 0.000 @w 2413.73 2227.10 m 2370.24 2226.89 L S @j 0.00 0.00 0.00 1.00 K 0.00 0.00 0.00 1.00 k 0 @g 0 @G [] 0 d 0 J 0 j 0 R 0 O 0 1.73 1.73 0 @w 2375.78 2222.50 m 2368.94 2226.89 L 2375.78 2231.35 L S @J @rs @rs @sm %EndColorLayer @rs @rs %EndPage %%Trailer @EndSysCorelDict end %%DocumentSuppliedResources: procset wCorel5Dict %%DocumentNeededResources: font Times-Roman %%+ font Times-Italic ---------------0009270459836 Content-Type: application/postscript; name="bmb2.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bmb2.eps" %!PS-Adobe-2.0 EPSF-2.0 %%BoundingBox: 2142 2153 2469 2455 %%Creator: CorelDRAW! %%Title: BMB2.EPS %%CreationDate: Mon Jul 12 22:24:27 1999 %%DocumentProcessColors: Black %%DocumentSuppliedResources: (atend) %%DocumentNeededResources: (atend) %%EndComments %%BeginSetup /AutoFlatness true def %Color profile: _DEFAULT.CCS - Generisch CMYK Farbe Drucker %%EndSetup %%BeginProlog %%BeginResource: procset wCorel5Dict % Copyright (c)1992-94 Corel Corporation % All rights reserved. v5.0 r0.6 /wCorel5Dict 300 dict def wCorel5Dict begin/bd{bind def}bind def/ld{load def} bd/xd{exch def}bd/_ null def/rp{{pop}repeat}bd/@cp/closepath ld/@gs/gsave ld /@gr/grestore ld/@np/newpath ld/Tl/translate ld/$sv 0 def/@sv{/$sv save def}bd /@rs{$sv restore}bd/spg/showpage ld/showpage{}bd currentscreen/@dsp xd/$dsp /@dsp def/$dsa xd/$dsf xd/$sdf false def/$SDF false def/$Scra 0 def/SetScr /setscreen ld/setscreen{3 rp}bd/@ss{2 index 0 eq{$dsf 3 1 roll 4 -1 roll pop} if exch $Scra add exch load SetScr}bd/SepMode where{pop}{/SepMode 0 def}ifelse /CurrentInkName where{pop}{/CurrentInkName(Composite)def}ifelse/$ink where {pop}{/$ink -1 def}ifelse/$c 0 def/$m 0 def/$y 0 def/$k 0 def/$t 1 def/$n _ def /$o 0 def/$fil 0 def/$C 0 def/$M 0 def/$Y 0 def/$K 0 def/$T 1 def/$N _ def/$O 0 def/$PF false def/s1c 0 def/s1m 0 def/s1y 0 def/s1k 0 def/s1t 0 def/s1n _ def /$bkg false def/SK 0 def/SM 0 def/SY 0 def/SC 0 def/$op false def matrix currentmatrix/$ctm xd/$ptm matrix def/$ttm matrix def/$stm matrix def/$fst 128 def/$pad 0 def/$rox 0 def/$roy 0 def/$ffpnt true def/CorelDrawReencodeVect[ 16#0/grave 16#5/breve 16#6/dotaccent 16#8/ring 16#A/hungarumlaut 16#B/ogonek 16#C/caron 16#D/dotlessi 16#27/quotesingle 16#60/grave 16#7C/bar 16#82/quotesinglbase/florin/quotedblbase/ellipsis/dagger/daggerdbl 16#88/circumflex/perthousand/Scaron/guilsinglleft/OE 16#91/quoteleft/quoteright/quotedblleft/quotedblright/bullet/endash/emdash 16#98/tilde/trademark/scaron/guilsinglright/oe 16#9F/Ydieresis 16#A1/exclamdown/cent/sterling/currency/yen/brokenbar/section 16#a8/dieresis/copyright/ordfeminine/guillemotleft/logicalnot/minus/registered/macron 16#b0/degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph/periodcentered 16#b8/cedilla/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters/questiondown 16#c0/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla 16#c8/Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis 16#d0/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply 16#d8/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls 16#e0/agrave/aacute/acircumflex/atilde/adieresis/aring/ae/ccedilla 16#e8/egrave/eacute/ecircumflex/edieresis/igrave/iacute/icircumflex/idieresis 16#f0/eth/ntilde/ograve/oacute/ocircumflex/otilde/odieresis/divide 16#f8/oslash/ugrave/uacute/ucircumflex/udieresis/yacute/thorn/ydieresis]def /@BeginSysCorelDict{systemdict/Corel10Dict known{systemdict/Corel10Dict get begin}if}bd/@EndSysCorelDict{systemdict/Corel10Dict known{end}if}bd AutoFlatness{/@ifl{dup currentflat exch sub 10 gt{ ([Error: PathTooComplex; OffendingCommand: AnyPaintingOperator]\n)print flush @np exit}{currentflat 2 add setflat}ifelse}bd/@fill/fill ld/fill{currentflat{ {@fill}stopped{@ifl}{exit}ifelse}bind loop setflat}bd/@eofill/eofill ld/eofill {currentflat{{@eofill}stopped{@ifl}{exit}ifelse}bind loop setflat}bd/@clip /clip ld/clip{currentflat{{@clip}stopped{@ifl}{exit}ifelse}bind loop setflat} bd/@eoclip/eoclip ld/eoclip{currentflat{{@eoclip}stopped{@ifl}{exit}ifelse} bind loop setflat}bd/@stroke/stroke ld/stroke{currentflat{{@stroke}stopped {@ifl}{exit}ifelse}bind loop setflat}bd}if/d/setdash ld/j/setlinejoin ld/J /setlinecap ld/M/setmiterlimit ld/w/setlinewidth ld/O{/$o xd}bd/R{/$O xd}bd/W /eoclip ld/c/curveto ld/C/c ld/l/lineto ld/L/l ld/rl/rlineto ld/m/moveto ld/n /newpath ld/N/newpath ld/P{11 rp}bd/u{}bd/U{}bd/A{pop}bd/q/@gs ld/Q/@gr ld/` {}bd/~{}bd/@{}bd/&{}bd/@j{@sv @np}bd/@J{@rs}bd/g{1 exch sub/$k xd/$c 0 def/$m 0 def/$y 0 def/$t 1 def/$n _ def/$fil 0 def}bd/G{1 sub neg/$K xd _ 1 0 0 0/$C xd /$M xd/$Y xd/$T xd/$N xd}bd/k{1 index type/stringtype eq{/$t xd/$n xd}{/$t 0 def/$n _ def}ifelse/$k xd/$y xd/$m xd/$c xd/$fil 0 def}bd/K{1 index type /stringtype eq{/$T xd/$N xd}{/$T 0 def/$N _ def}ifelse/$K xd/$Y xd/$M xd/$C xd }bd/x/k ld/X/K ld/sf{1 index type/stringtype eq{/s1t xd/s1n xd}{/s1t 0 def/s1n _ def}ifelse/s1k xd/s1y xd/s1m xd/s1c xd}bd/i{dup 0 ne{setflat}{pop}ifelse}bd /v{4 -2 roll 2 copy 6 -2 roll c}bd/V/v ld/y{2 copy c}bd/Y/y ld/@w{matrix rotate /$ptm xd matrix scale $ptm dup concatmatrix/$ptm xd 1 eq{$ptm exch dup concatmatrix/$ptm xd}if 1 w}bd/@g{1 eq dup/$sdf xd{/$scp xd/$sca xd/$scf xd}if }bd/@G{1 eq dup/$SDF xd{/$SCP xd/$SCA xd/$SCF xd}if}bd/@D{2 index 0 eq{$dsf 3 1 roll 4 -1 roll pop}if 3 copy exch $Scra add exch load SetScr/$dsp xd/$dsa xd /$dsf xd}bd/$ngx{$SDF{$SCF SepMode 0 eq{$SCA}{$dsa}ifelse $SCP @ss}if}bd/p{ /$pm xd 7 rp/$pyf xd/$pxf xd/$pn xd/$fil 1 def}bd/@MN{2 copy le{pop}{exch pop} ifelse}bd/@MX{2 copy ge{pop}{exch pop}ifelse}bd/InRange{3 -1 roll @MN @MX}bd /wDstChck{2 1 roll dup 3 -1 roll eq{1 add}if}bd/@dot{dup mul exch dup mul add 1 exch sub}bd/@lin{exch pop abs 1 exch sub}bd/cmyk2rgb{3{dup 5 -1 roll add 1 exch sub dup 0 lt{pop 0}if exch}repeat pop}bd/rgb2cmyk{3{1 exch sub 3 1 roll}repeat 3 copy @MN @MN 3{dup 5 -1 roll sub neg exch}repeat}bd/rgb2g{2 index .299 mul 2 index .587 mul add 1 index .114 mul add 4 1 roll 3 rp}bd/WaldoColor where{pop} {/SetRgb/setrgbcolor ld/GetRgb/currentrgbcolor ld/SetGry/setgray ld/GetGry /currentgray ld/SetRgb2 systemdict/setrgbcolor get def/GetRgb2 systemdict /currentrgbcolor get def/SetHsb systemdict/sethsbcolor get def/GetHsb systemdict/currenthsbcolor get def/rgb2hsb{SetRgb2 GetHsb}bd/hsb2rgb{3 -1 roll dup floor sub 3 1 roll SetHsb GetRgb2}bd/setcmykcolor where{pop/SetCmyk /setcmykcolor ld}{/SetCmyk{cmyk2rgb SetRgb}bd}ifelse/currentcmykcolor where{ pop/GetCmyk/currentcmykcolor ld}{/GetCmyk{GetRgb rgb2cmyk}bd}ifelse /setoverprint where{pop}{/setoverprint{/$op xd}bd}ifelse/currentoverprint where {pop}{/currentoverprint{$op}bd}ifelse/@tc{5 -1 roll dup 1 ge{pop}{4{dup 6 -1 roll mul exch}repeat pop}ifelse}bd/@trp{exch pop 5 1 roll @tc}bd /setprocesscolor{SepMode 0 eq{SetCmyk}{0 4 $ink sub index exch pop 5 1 roll 4 rp SepsColor true eq{$ink 3 gt{1 sub neg SetGry}{0 0 0 4 $ink roll SetCmyk} ifelse}{1 sub neg SetGry}ifelse}ifelse}bd/findcmykcustomcolor where{pop}{ /findcmykcustomcolor{5 array astore}bd}ifelse/setcustomcolor where{pop}{ /setcustomcolor{exch aload pop SepMode 0 eq{pop @tc setprocesscolor}{ CurrentInkName eq{4 index}{0}ifelse 6 1 roll 5 rp 1 sub neg SetGry}ifelse}bd} ifelse/@scc{dup type/booleantype eq{setoverprint}{1 eq setoverprint}ifelse dup _ eq{pop setprocesscolor pop}{findcmykcustomcolor exch setcustomcolor}ifelse SepMode 0 eq{true}{GetGry 1 eq currentoverprint and not}ifelse}bd/colorimage where{pop/ColorImage/colorimage ld}{/ColorImage{/ncolors xd pop/dataaq xd{ dataaq ncolors dup 3 eq{/$dat xd 0 1 $dat length 3 div 1 sub{dup 3 mul $dat 1 index get 255 div $dat 2 index 1 add get 255 div $dat 3 index 2 add get 255 div rgb2g 255 mul cvi exch pop $dat 3 1 roll put}for $dat 0 $dat length 3 idiv getinterval pop}{4 eq{/$dat xd 0 1 $dat length 4 div 1 sub{dup 4 mul $dat 1 index get 255 div $dat 2 index 1 add get 255 div $dat 3 index 2 add get 255 div $dat 4 index 3 add get 255 div cmyk2rgb rgb2g 255 mul cvi exch pop $dat 3 1 roll put}for $dat 0 $dat length ncolors idiv getinterval}if}ifelse}image}bd }ifelse/setcmykcolor{1 5 1 roll _ currentoverprint @scc/$ffpnt xd}bd /currentcmykcolor{0 0 0 0}bd/setrgbcolor{rgb2cmyk setcmykcolor}bd /currentrgbcolor{currentcmykcolor cmyk2rgb}bd/sethsbcolor{hsb2rgb setrgbcolor} bd/currenthsbcolor{currentrgbcolor rgb2hsb}bd/setgray{dup dup setrgbcolor}bd /currentgray{currentrgbcolor rgb2g}bd}ifelse/WaldoColor true def/@sft{$tllx $pxf add dup $tllx gt{$pwid sub}if/$tx xd $tury $pyf sub dup $tury lt{$phei add}if/$ty xd}bd/@stb{pathbbox/$ury xd/$urx xd/$lly xd/$llx xd}bd/@ep{{cvx exec }forall}bd/@tp{@sv/$in true def 2 copy dup $lly le{/$in false def}if $phei sub $ury ge{/$in false def}if dup $urx ge{/$in false def}if $pwid add $llx le{/$in false def}if $in{@np 2 copy m $pwid 0 rl 0 $phei neg rl $pwid neg 0 rl 0 $phei rl clip @np $pn cvlit load aload pop 7 -1 roll 5 index sub 7 -1 roll 3 index sub Tl matrix currentmatrix/$ctm xd @ep 4 rp}{2 rp}ifelse @rs}bd/@th{@sft 0 1 $tly 1 sub{dup $psx mul $tx add{dup $llx gt{$pwid sub}{exit}ifelse}loop exch $phei mul $ty exch sub 0 1 $tlx 1 sub{$pwid mul 3 copy 3 -1 roll add exch @tp pop}for 2 rp}for}bd/@tv{@sft 0 1 $tlx 1 sub{dup $pwid mul $tx add exch $psy mul $ty exch sub{dup $ury lt{$phei add}{exit}ifelse}loop 0 1 $tly 1 sub{$phei mul 3 copy sub @tp pop}for 2 rp}for}bd/@pf{@gs $ctm setmatrix $pm concat @stb eoclip Bburx Bbury $pm itransform/$tury xd/$turx xd Bbllx Bblly $pm itransform/$tlly xd/$tllx xd/$wid $turx $tllx sub def/$hei $tury $tlly sub def @gs $vectpat{1 0 0 0 0 _ $o @scc{eofill}if}{$t $c $m $y $k $n $o @scc{SepMode 0 eq $pfrg or{ $tllx $tlly Tl $wid $hei scale <00> 8 1 false[8 0 0 1 0 0]{}imagemask}{/$bkg true def}ifelse}if}ifelse @gr $wid 0 gt $hei 0 gt and{$pn cvlit load aload pop /$pd xd 3 -1 roll sub/$phei xd exch sub/$pwid xd $wid $pwid div ceiling 1 add /$tlx xd $hei $phei div ceiling 1 add/$tly xd $psx 0 eq{@tv}{@th}ifelse}if @gr @np/$bkg false def}bd/@dlt{$fse $fss sub/nff xd $frb dup 1 eq exch 2 eq or{ $frt dup $frc $frm $fry $frk @tc 4 copy cmyk2rgb rgb2hsb 3 copy/myb xd/mys xd /myh xd $tot $toc $tom $toy $tok @tc cmyk2rgb rgb2hsb 3 1 roll 4 1 roll 5 1 roll sub neg nff div/kdb xd sub neg nff div/kds xd sub neg dup 0 eq{pop $frb 2 eq{.99}{-.99}ifelse}if dup $frb 2 eq exch 0 lt and{1 add}if dup $frb 1 eq exch 0 gt and{1 sub}if nff div/kdh xd}{$frt dup $frc $frm $fry $frk @tc 5 copy $tot dup $toc $tom $toy $tok @tc 5 1 roll 6 1 roll 7 1 roll 8 1 roll 9 1 roll sub neg nff dup 1 gt{1 sub}if div/$dk xd sub neg nff dup 1 gt{1 sub}if div/$dy xd sub neg nff dup 1 gt{1 sub}if div/$dm xd sub neg nff dup 1 gt{1 sub}if div/$dc xd sub neg nff dup 1 gt{1 sub}if div/$dt xd}ifelse}bd/ffcol{5 copy $fsit 0 eq{ setcmykcolor pop}{SepMode 0 ne{$frn findcmykcustomcolor exch setcustomcolor}{4 rp $frc $frm $fry $frk $frn findcmykcustomcolor exch setcustomcolor}ifelse} ifelse}bd/@ftl{1 index 4 index sub dup $pad mul dup/$pdw xd 2 mul sub $fst div /$wid xd 2 index sub/$hei xd pop Tl @dlt $fss 0 eq{ffcol n 0 0 m 0 $hei l $pdw $hei l $pdw 0 l @cp $ffpnt{fill}{@np}ifelse}if $fss $wid mul $pdw add 0 Tl nff {ffcol n 0 0 m 0 $hei l $wid $hei l $wid 0 l @cp $ffpnt{fill}{@np}ifelse $wid 0 Tl $frb dup 1 eq exch 2 eq or{4 rp myh mys myb kdb add 3 1 roll kds add 3 1 roll kdh add 3 1 roll 3 copy/myb xd/mys xd/myh xd hsb2rgb rgb2cmyk}{$dk add 5 1 roll $dy add 5 1 roll $dm add 5 1 roll $dc add 5 1 roll $dt add 5 1 roll} ifelse}repeat 5 rp $tot dup $toc $tom $toy $tok @tc ffcol n 0 0 m 0 $hei l $pdw $hei l $pdw 0 l @cp $ffpnt{fill}{@np}ifelse 5 rp}bd/@ftrs{1 index 4 index sub dup $rox mul/$row xd 2 div 1 index 4 index sub dup $roy mul/$roh xd 2 div 2 copy dup mul exch dup mul add sqrt $row dup mul $roh dup mul add 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