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\begin{document}

\title[A Generalization of Sard's Theorem]{A Generalization of Sard's Theorem}

\author{S.M. Bates and C.G. Moreira}

\address{Department of Mathematics, Columbia University, New York, NY, 10027, USA}

\address{IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, RJ, Brasil}

\email{smb@@math.columbia.edu}

\email{gugu@@impa.br}

\thanks{Supported by an NSF Postdoctoral Fellowship in Mathematics.}

\thanks{Supported by CNPq, Brasil.}

\maketitle

\thispagestyle{empty}

\baselineskip=15pt

\begin{abstract}
Let $E$ be a euclidean space and $F$ any normed vector space. If $A \subset E$ is a $t$-sigmafinite
set of rank $r$ for a $C^k$ function $f\! : E \ra F$, then $f(A)$ is $(r + (t-r)/k)$-null for any $t>r$.
We also prove that if $\ga \in (0,1]$ and $A\subset E$ is a $t$-sigmafinite (resp.\ $t$-null) set of rank
$r$ for a $C^{k+\ga}$ mapping $f\! : E \ra F$, then $f(A)$ is $(r + (t-r)/k)$-sigmafinite (resp.\ $(r + (t-r)/k)$-null)
for any $t>r$.

If $E$ is an infinite-dimensional Banach space, then the theorems above are true for $k\leq 2$. We give examples of
polynomial mappings that show that we cannot improve our results by improving the differentiability hypothesis. In particular,
we exhibit a polynomial $P\! : \ell _2 \ra \reals$ which maps a 3-finite critical set onto the interval $[0,1]$.

\end{abstract}


\section*{Introduction}

For $t\in \integers _+$, we recall that a set $A\subset \rn$ is said to be (countably) $t$-rectifiable provided all but at 
most a $t$-null subset of $A$ is contained in a countable union of $C^1$-embedded $t$-dimensional submanifolds of $\rn$.

\begin{thm2}
\label{thm-main1}

Fix $t,s >0$ and $r \in \naturals$ with $t > r$ and set $d = r + (t-r)/s$. Let $E$ be a euclidean space and $F$ any
normed vector space. Let $A \subset E$ be a $t$-sigmafinite set of rank r for a differentiable map $f\! : E \ra F$.

\ben

\item If $s \in \integers _+$ and $f \in C^s$, then $f(A)$ is $d$-null.
\item If $s > 1$ and $f \in \gL _s$, then $\calh ^d(f(A)) \leq (2 \| f \| _s)^d (1+r)^{d/2}\calh ^t (A).$
\item If $f \in \gL _s$, $t \in \integers _+$ and $A$ is $t$-rectifiable, then $f(A)$ is $d$-null.

\een

\end{thm2}

Note that if $k>1$ and $t<r$, then $t< r + (t - r)/k$, and the conclusion of the theorem in this case
follows trivially from the fact that any $C^1$ function maps $t$-sigmafinite sets to $t$-sigmafinite sets.
If $t=r$, then the statement of the theorem is false, as illustrated by the identity map on $\reals ^r$.
Finally, we note that if $F$ is finite-dimensional and $r = \dim (F)$, then the assertion follows from
then inequality $r < r + (t-r)/k$, since $F$ itself is $(r + (t-r)/k)$-null. 

For differentiable mappings from infinite-dimensional domains, similar results hold in the smoothness
classes $\gL _s$ for $s \in (1,3]$. When $t= \dim (E)$, statement (2) subsumes previous results of 
\cite{bates, moreira, norton}.

\begin{thm2}
\label{thm-main2}

Fix $s \in [1,3]$ and define $d = r + (t-r)/s$. Suppose $X,Y$ are Banach spaces and $A\subset X$ is a $t$-sigmafinite
rank-$r$ set for a differentiable map $f \! : X \ra Y$.

\ben

\item If $s \in \{ 1,2 \}$ and $f$ is $C^s$, then $f(A)$ is $d$-null.
\item If $f \in \gL _s$, then $\calh ^d(f(A)) \leq (2 \| f \| _s)^d (1+r)^{d/2}\calh ^t (A).$
\item If $f \in \gL _s$, $t \in \integers _+$ and $A$ is $t$-rectifiable, then $f(A)$ is $d$-null.

\een

\end{thm2}
In particular, Theorem 2 asserts that for any Banach space $X$, a $C^3$ real-valued function maps 
any $t$-dimensional critical set to a set of dimension $\leq 3$.

\begin{thm2}
\label{thm-main3}

For any infinite-dimensional Banach space $X$, there exists a cubic polynomial $P\! :X \ra \reals$ and
a 3-finite critical set $A\subset X$ such that $P(A) = [0,1]$.

\end{thm2}

\vspace{.1in}

\nin {\em Notations and Definitions.}

For $s\in (0,1)$, we denote by $\gL _s$ the Lipschitz space of those continuous functions $f\! : \reals \ra \reals$
for which 

\[ \| f\| _s = \sup_{x\in \reals} |f(x)| + \sup_{x,h\in \reals} \frac{| f(x+h) - f(x)|}{|h|^s} < \infty. \]
A function $f \! : \reals \ra \reals$ belongs to $\gL _1$ if $f$ is continuous and 
\[ \| f\| _1 = \sup_{x\in \reals} |f(x)| + \sup_{x,h\in \reals} \frac{| f(x+h) + f(x-h) - 2f(x)|}{|h|^s} < \infty. \]
For $s > 1$, let $[s]$ denote the greatest integer {\em strictly less than} $s$. The class $\gL _s$ is then defined
inductively as the space of $C^{[s]}$ functions $f \! : \reals \ra \reals$ for which 
\[ \| f \| _s = \| f \|_{s-1} + \| f' \| _{s-1} < \infty . \]
When $s >1 $ is not an integer, each element of the H\"older class $C^s$ coincides locally with some member of 
$\gL _s$. For $k \in \integers _+$, the Lipschitz space $\gL _k$ strictly contains $C^k$. A function $f\in C^{k-1}$
coincides locally with some member of $\gL _k$ provided that the $(k-1)$-st derivative of $f$ satisfies a local 
Zygmund condition (see \cite{stein}).

\section{Decompositions of Critical Sets}

\begin{lemma}
\label{lemma-main}
Fix $\gh > 0$. Let $E$ be a euclidean space, $F$ a normed vector space, and $A\subset E$ a set of rank-$r$ points
for a differentiable map $f \! : E \ra F$.

\ben

\item If $s \in \integers _+$, $s \geq 2$, and $f \in C^s$, then there exists a decomposition $\{ A_i \}$ 
of $A$ such that for each $i$, there is a constant $M_i > 0$ and a splitting of $E$ such that for any set
$S \subset E$,
\[ | f(S\cap A_i)| \leq (\| f \| _1 + \gh ) |S'| + M_i |S'||S''| + \gh |S|^s. \]

\item If $s > 2$ and $f \in \gL _s$, then there exists a decomposition $\{ A_i \}$ of $A$ such that for 
each $i$, there is a constant $M_i > 0$ and a splitting of $E$ such that for any set $S \subset E$,
\[ |f(S\cap A_i)| \leq (\| f\| _1 +\gh)|S'| + M_i |S'||S''| + M_i |S|^s. \]

\item If $A$ is $t$-rectifiable for some $t \in \integers _+$, then there exists a decomposition $\{ A_i \}$ of $A$
such that $A_0$ is $t$-null, $\sum_i |A_i|^t \leq \calh ^t (A_i) + 1$, and for each $i\in \integers _+$, there is a 
constant $M_i > 0$ and a splitting of $E$ such that for any set $S\subset E$,
\[ |f(S \cap A_i)| \leq (\| f\| _1 + \gh )|S'| + M_i |S'||A_i| + M_i |A_i|^s. \]

\een

\end{lemma}
To prove the lemma, we will need the following generalization of the Morse decomposition lemma:

\begin{lemma}
\label{lemma-main2}
Let $E$ be a euclidean space, $F$ any normed vector space, and $h\! : E \ra F$ a $C^k$ map. If $h=0$ on $A\subset E$,
then for each $\gh >0$ there exists a decomposition $\{ A_i \}$ of $A$ such that $A_0$ is countable, and for each
$i \in \integers _+$, there is a closed ball $B_i \subset \reals ^{n_i}$ and a $C^1$ map $\gps _i \! : B_i \ra E$ such
that 

\ben

\item $A_i \subset \gps _i (B_i)$ and $| x-y | \leq |\gps _i (x) - \gps _i (y) | \leq (1+\gh ) |x-y|$ for all $x,y \in B_i$.

\item For all $x \in \gps _i^{-1} (A_i)$ and $y \in B_i$,
\[ h(\gps _i (y)) - h(\gps _i (x))| \leq \gh |x-y|^k. \]

\een

\end{lemma}

\pff Let $(n,k)$ denote the statement of the lemma for all $\gh >0$
and any $C^k$ mapping from $n$-dimensional euclidean space into any
Banach space.

We first prove $(1,k)$ for all $k\in \integers _+$. Fix $\gh >0$ and
let $A_0$ be the set of isolated points of $A$. If $\dim (E) = 1$ and
$F$ is any Banach space, then $D^i \! h(x) =0$ for each $i=1, \cdots ,
k$ and $x\in A \setminus A_0$. (This follows easily from the mean-value
theorem when $F = \reals$ and can in general be deduced from the
equation $D^i(y^* \circ f) = y^* \circ D^i \! f$, valid for each
$y^* \in F^*$). By continuity of $D^k \! f$, we can choose a covering
of $A\setminus A_0$ by closed intervals $B_i$ such that $|D^k \! h|
< \gh$ on each $B_i$. From Taylor's theorem, it follows that for each
$x\in A_i$ and $y\in B_i$,
\[ |h(y)| = |h(y) - h(x)| \leq \gh |x-y|^k, \]
which proves statement $(1,k)$.

To prove case $(n,k)$ of the lemma, we proceed inductively, assuming
that statements $(n-1,k)$ and $(n,k-1)$ have been proven (if $k=1$,
we only assume $(n-1,1)$). Given a subset $A$ of an $n$-dimensional
euclidean space $E$, define
\[ U_A = \{ x\in A : \mbox{every $C^k$ function $g\! : E \ra \reals$
with $g(A) = 0$ satisfies $D\! g (x) = 0$} \}, \]
$V_A = A \setminus U_A$. Fix $\gh >0$. To decompose $A$, we find
appropriate decompositions of $V_A$ and $U_A$.

>From the definition of $V_A$, it follows that for each $\gh >0$,
there exists a countable collection of $C^1$ embeddings 
$\gvf _i \! : B_i' \ra E$ such that each $B_i'$ is a closed
ball in $\reals ^{n-1}$, $V_A \subset \bigcup_i \gvf _i (B_i')$,
and 
\[ |x-y| \leq |\gvf _i (x) - \gvf _i (y)| \leq (1+\gh )^{1/2} |x-y|. \]
By applying the $(n-1,k)$ statement of the lemma to the mapping
$h \circ \gvf _i$, we obtain a decomposition $\{ A_i' \}$ of 
$\gvf _i^{-1}(V_A)$ and a sequence of $C^1$ maps 
$\gps _{ij} \! : B_{ij} \ra B_i$ such that:

\ben

\item $A_j' \subset \gps _{ij}(B_{ij})$ and $|x-y| \leq
|\gps _{ij}(x) - \gps_{ij}(y)| \leq (1+\gh )^{1/2}|x-y|$ for
all $x,y \in B_{ij}$.

\item For all $y \in B_{ij}$,
\[ | h(\gvf _i (\gps _{ij}(y)))| \leq \gh d(y,\gps _{ij}^{-1}
(A_j'))^k. \]

\een

\nin Setting $V_{ij} = \gvf _i(A_j')$ and $\tilde{\gps}_{ij} = 
\gvf _i \circ \gps _{ij}$ then gives the desired decomposition
of $V_A$.

Next, we note that $Dh = 0$ on $U_A$. If $k=1$, then we can use
the continuity of $Dh$ to find a countable covering of $U_A$ by
balls $B_i$ on which $\| Dh \| \leq \gh$, so that $\{ B_i \cap
U_A \}$ gives the desired decomposition of $U_A$. For $k>1$, we
apply the $(n,k-1)$ hypothesis to the map $Dh \! : E \ra L(E,F)$
in order to find a decomposition $\{ U_i \}$ of $U_A$ and mappings
$\gps _i \! : B_i \ra E$ such that 

\ben

\item $U_i \subset \gps _i (B_i)$ and $|x-y| \leq |\gps _i(x)
- \gps _i(y) | \leq (1+\gh )|x-y|$ for all $x,y \in B_i$.

\item For all $x\in \gps _i^{-1}(U_i)$ and $y\in B_i$,
\[ | Dh(\gps _i(y)) - Dh(\gps _i(x))| \leq \gh |x-y|^{k-1}. \]

\een

\nin Given $x \in \gps _i^{-1}(U_i)$ and $y\in B_i$, we set 
$\gc (t) = ty + (1-t)x$, so that
\begin{eqnarray*}
|h(\gps _i(y))| & = & \left| \int_0^1 D(h\circ \gps _i)(\gc (t))
\gc '(t) \, dt \right| \\
& \leq & \gh |x-y|^{k-1} |x-y|.
\end{eqnarray*}
Thus, $|h(\gps _i(y))| \leq \gh d(y,\gps _i^{-1}(U_i))^k$ for all
$y\in B_i$, and this completes the proof.
\q

\begin{lemma}
\label{lemma-main3}
Let $E$ be a euclidean space, $F$ any normed vector space, and 
$h\! : E \ra F$ a $\gL _s$ map with $s>2$. If $h=0$ on $A\subset E$,
then for each $\gh >0$ there exists a decomposition $\{ A_i \}$ of $A$ 
such that $A_0$ is countable, and for each $i \in \integers _+$, there 
is a closed ball $B_i \subset \reals ^{n_i}$ and a $C^1$ map 
$\gps _i \! : B_i \ra E$ such that 

\ben

\item $A_i \subset \gps _i (B_i)$ and $| x-y | \leq 
|\gps _i (x) - \gps _i (y) | \leq (1+\gh ) |x-y|$ for all $x,y \in B_i$.

\item For all $x, x' \in \gps _i^{-1} (A_i)$ and $y$ in the line segment
$xx'$,
\[ |h(\gps _i (y)) - h(\gps _i (x))| \leq M_i |x-y|^{s-1} |x-x'|. \]

\een

\end{lemma}

\vspace{.1in}

\nin {\bf Proof of Lemma~\ref{lemma-main}.} We first prove the lemma
assuming that with respect to some splitting $E = E' \oplus E''$, the
map $f$ is of the form
\[ f(z,y) = (h(z),g(z,y)) \]
for some $h,g$ such that $D_2g = 0$ on $A$. From Lemma~\ref{lemma-main2}
we obtain a decomposition $\{ A_i \}$ of $A$ and $C^1$ mappings 
$\gps _i \! : B_i \ra E$ such that

\ben

\item $A_i \subset \gps_i (B_i)$ and $|x-y| \leq |\gps_i (x) -
\gps_i (y)| \leq (1+\gh )|x-y|$ for all $x,y \in B_i$.

\item For all $x,x' \in \gps _i^{-1} (A_i)$ and $y\in xx'$,
\[ | D_2g(\gps _i (y)) | \leq \gh |x-x'|^{s-1}. \]

\een

\nin Fix $i\in \integers _+$ and suppose that $(z,y),(z',y') 
\in S \cap A_i$. Then

\begin{eqnarray*}
|f(z,y) - f(z',y') | & \leq & |f(z,y) - f(z',y)| + |f(z',y) - f(z',y')| \\
& \leq & \| f \| _1 |S'| + |g(z',y) - g(z',y')| . 
\end{eqnarray*}

\nin To bound the term on the right, let $\gc \! : [0,1] \ra E$ be
the composition of $\gps _i$ with the affine path between $\gps _i^{-1}
(z,y)$ and $\gps _i^{-1}(z',y')$. If $\tilde{\gc} \! : [0,1] \ra E$
is the composition of $\gc$ with the projection onto $z'\times E''$,
then 

\begin{eqnarray*}
| g(z,y) - g(z',y') | & = & \left| \int_0^1 D_2g (\tilde{\gc}(t)) 
\tilde{\gc}'(t) \, dt \right| \\
& \leq & \int_0^1 | D_2g( \tilde{\gc}(t)) - D_2g( \tilde{\gc}(t)) |
|\tilde{\gc}' (t)| \, dt + \int_0^1 |D_2g(\gc (t))| |\tilde{\gc}' (t)| 
\, dt. 
\end{eqnarray*}

\nin Since $|\tilde{\gc}'(t) | \leq (1 + \gh ) |S''|$ for all $t$, we
have 
\[ | D_2g( \tilde{\gc}(t)) - D_2g( \tilde{\gc}(t)) | |\tilde{\gc}' (t)|
\leq (1 + \gh ) \| f \| _2 |S'||S''|. \]
Finally, if $x,y \in B_i$ satisfy $\gps _i (x), \gps _i(y) \in S \cap A_i$
and $\gps _i(y) \in Q$, then property (1) of the map $\gps _i$ implies
that $|x-y| \leq |S|$. From (2), it follows that $\| D_2g \|$ is bounded
on $S \cap \gps_i (B_i)$ by $\gh |S|^{s-1}$, and so
\[ \int_0^1 \| D_2g (\gc (t)) \| | \tilde{\gc}'(t) | \, dt \leq (1+ \gh )
\gh |S|^s . \]
>From these estimates, part (1) of the lemma follows.

\vspace{.1in}

Part (2) of the lemma follows by exactly the same argument, where
Lemma~\ref{lemma-main3} is applied in place of Lemma~\ref{lemma-main2}
to decompose $A$.

\vspace{.1in}

Turning to part (3) of the lemma, we suppose that $A$ is $t$-rectifiable.
By combining Lemma~\ref{lemma-main3} with the definition of rectifiability,
we then obtain a countable decomposition $\{ A_i \}$ of $A$ such that
$A_0$ is $t$-null, and for each $i \in \integers _+$, there exists a
ball $B_i \subset \reals ^t$ and a $C^1$ embedding $\gps _i \! : B_i \ra E$
such that 

\ben

\item $A_i \subset \gps _i (B_i)$ and each point of $\gps _i^{-1}(A_i)$
is a Lebesgue density point.

\item $|x-y| \leq |\gps _i(x) - \gps _i(y) | \leq 2|x-y|$ for all 
$x,y \in B_i$.

\item For all $x, x' \in \gps _i^{-1}(A_i)$ and $y\in xx'$,
\[ |D_2g(\gps _i (y))| \leq M_i |x-x'|^{s-1}. \]

\een

\nin To obtain the desired decomposition of $A$, we will further 
decompose each $A_i$. Given $\gh >0$, we fix a positive integer $P$
such that $2/P < \gh$. Since each $x\in \gps _i^{-1}(A_i)$ is a 
density point, there exists $\gel _x > 0$ such that if $Q(x,\gel )$
is a cube of edge $\gel < \gel _x$ centered at $x \in \gps _i^{-1}
(A_i)$, then
$$ \frac{\calh ^t(Q\cap \gps _i^{-1} (A_i))}{\calh ^t(Q)} \geq
1 - P^{-t}. \eqno{(*)} $$
The collection
\[ \calq = \{ Q(x, \gel ) : x\in \gps _i^{-1} (A_i), \gel < \gel _x \} \]
is a Vitali family for $\gps _i^{-1}(A_i)$, and so there exists a 
sequence of pairwise disjoint cubes $\{ Q_j \} \subset \calq$ such
that $\gps _i^{-1}(A_i) \setminus \bigcup_j Q(x_j, \gel _j)$ is 
$t$-null and $\sum_j \gel _j^t < \calh ^t(\gps _i^{-1}(A_i)) + 1$.

Fix $j$ and consider $(z,y), (z',y') \in Q_j \cap \gps _i^{-1}(A_i)$.
By condition $(*)$, it follows that there are $x_0, \cdots , x_P$ in 
$Q \cap \gps _i^{-1}(A_i)$ such that $x = x_0$, $y = x_P$, and
$|x_i - x_{i+1}| < 2|Q| P^{-1}$ for all $i$. Let $\gc \! : [0,1] \ra E$
be the composition of $\gps _i$ with the piecewise linear path 
connecting the points $x_i$. If $\tilde{\gc} \! : [0,1] \ra E$ is the
composition of $\gc$ with the projection onto $z'\times E''$, then

\begin{eqnarray*}
|g(z,y) - g(z',y')| & = & \left| \int_0^1 D_2g(\tilde{\gc}(t)) 
\tilde{\gc} ' (t) \, dt \right| \\
& \leq & \int_0^1 |D_2g(\tilde{\gc}(t)) - D_2g(\gc(t))| 
|\tilde{\gc} ' (t)| \, dt + \int_0^1 |D_2g(\gc (t))| |\tilde{\gc}'(t)| 
\, dt. 
\end{eqnarray*}

\nin Since $|\tilde{\gc}(t)| \leq 2|Q_j|$ for all $t$, we have
\[ |D_2g(\tilde{\gc (t)}) - D_2(\gc (t))| |\tilde{\gc}(t)| \leq
2\| f\| _2 |S'| |Q_j|. \]
Finally, property (1) of the map $\gps _i$ implies that for each $i$,
we have $|x_i - x_{i+1}| \leq 2|Q|/P$, and so from (3) it follows that
\[ \int_0^1 \| D_2g (\gc (t)) \| |\tilde{\gc}'(t)| \, dt \leq
2|Q_j| M_i \sum_{i=0}^{P-1}|x_i - x_{i+1}|^{s-1} \leq 2^s M_i P^{2-s}
|Q_j|^s. \]
Thus, proceeding as above, and applying the inequality 
$2^sM_i P^{2-s} < \gh$, we obtain
\[ |f(S\cap \gps_i (Q_j))| \leq \| f\| _1 |S'| + M_i|S'||Q_j| + 
\gh |Q_j|^s. \]

For each $i \in \integers _+$, we now choose a collection of cubes
$Q_j$ associated to $A_i$ as above. The decomposition $\{ A_i \}$
given by a denumeration of the sets $\gps _i(Q_j) \cap A$, and 
$A_0 = A \setminus \bigcup A_i$ then satisfies the lemma.

For the general case, we apply the implicit function theorem
locally to find a diffeomorphism $\gvf \! : E'\oplus E'' \ra E$
such that $f\circ \gvf$ is of the split form above. 
If $\gps \! : B \ra E$ is a map such that 
\[ | f(\gps _i (y)) | \leq \gh d(y,\gps _i^{-1}(A_i))^k \]
and $\gvf (x), \gvf (y) \in \gps (B)$, then....

Note that if we compose the map $f$ in Lemma~\ref{lemma-main3}
with a $C^1$ diffeomorphism $\gvf \! : E \ra E$, then we obtain
the estimate
\[ |(f\circ \gvf )(x) - (f \circ \gvf )(y) | \leq (\| f \| _s + \gh )
|\gvf (x) - \gvf (y)|^s \leq (\| f\| _s + \gh ) \| \gvf \| _1^s |x-y|^s. \]
In other words, we can compose Morse decompositions with diffeomorphism
$\gvf$ and only increase the constants by the $C^1$ norm of $\gvf$.
\q


\begin{lemma}
\label{lemma-main4}
Fix $\gh >0$. Let $E,F$ be Banach spaces and $A\subset E$ a set of
rank-$r$ points for a differentiable map $f\! : E \ra F$.

\ben

\item If $s\in \{ 1,2 \}$ and $f\in C^s$, then there exists a
decomposition $\{ A_i \}$ of $A$ such that for each $i$, there
is a splitting of $E$ such that for any set of the form $Q=Q' \times S''$
\[ | f(Q\cap A_i)| \leq \| f\| _1|Q'| + \gh |Q|^s. \]

\item If $s \in (1,3]$ and $f\in \gL _s$, then there exists a 
decomposition $\{ A_i \}$ of $A$ such that for each $i$, there is a 
splitting of $E$ such that for any set of the form $Q = Q' \times S''$
\[ | f(Q\cap A_i ) | \leq \| f\| _1|Q'| + \| f\| _s |Q|^s. \]

\een

\end{lemma}

\pff We first prove the lemma assuming that with respect to some 
fixed splitting $E' \oplus E''$ of $E$, the map $f$ assumes the form
\[ f(z,y) = (h(z),g(z,y)) \]
and $D_2g = 0$ on $A$. Fix $(z,y), (z',y') \in A$ and define
\[ \gc (t) = (t(z' - z) + z, t(y' - y) +y). \]
By the mean-value theorem, there exists a $c \in (0,1)$ such that

\begin{eqnarray*}
|f(z,y) - f(z',y')| & = & |D_1f(\gc (c)) (z'-z) + D_2f(\gc (c))(y'-y)| \\
& \leq & \| f \|_1 |Q'| + |D_2g (\gc (c))| |Q|, 
\end{eqnarray*}

\nin since $D_2f = D_2g$.

1. Let $\{ B_i \}$ be a countable covering of $A$ by balls such that
$|D_2 g| < \gh$ on each $B_i$, and set $A_i = A\cap B_i$. For sets of
the form $Q = Q' \times S''$, the preceding inequality then implies
that 
\[ |f(Q \cap A_i)| \leq \| f\| _1 |Q'| + \gh |Q|, \]
thus proving statement (1) of the lemma for the $C^1$ case. If $f\in C^2$,
then the mean-value theorem implies that for some $c' \in (0,c)$,
\[ |D_2g(\gc (c)) - D_2g(\gc (0))| \leq |D(D_2g)(\gc (c'))(z-z',y-y')|. \]
Since $D_2g(\gc (0)) = D_2g(\gc (1)) = 0$, there exists $c'' \in (0,1)$
such that $D(D_2g)(\gc (c''))(z-z',y-y') = 0$, so that
\[ |D_2g(\gc (c))| \leq |D(D_2g)(\gc (c')) - D(D_2g)(\gc (c'))|
|(z-z',y-y')|. \]
Let $\{ B_i \}$ be a covering of $A$ by balls $B_i$ such that
$|D(D_2g)(p) - D(D_2g)(p')| < \gh$ for all $p,p' \in B_i$, and 
set $A_i = A\cap B_i$. Combining $(*)$ with the preceding inequalities,
we find that for any set of the form $Q = Q' \times S''$,
\[ |f(Q\cap A_i)| \leq \| f\| _1 + \gh |Q|^2. \]

\vspace{.1in}

2. Turning to statement (2) of the lemma, we recall that in the
notation above, $D_2g(\gc (0)) = D_2g(\gc (1)) = 0$. If $f\in
\gL _s$ for $s\in (1,2]$, then Norton's lemma implies that
\[ |D_2g (\gc (c))| \leq \| f\| _s |Q|^{s-1}, \]
by Lemma~\ref{lemma-main5}. Combined with the preceding inequality,
this proves the lemma for $s \in (1,2]$.

For $s\in (2,3]$, fix $\gd >0$ and choose $q,q' \in A$ such that 
$|p-q|, |p'-q'| < \gd$. Let $\overline{\gc} \! : [-2\gd, 2\gd ] \ra
E$ be a smooth extension of $\gc$ such that $\| \gc \| _1 < |Q|$
and $\overline{\gc}(-2\gd ) =q$, $\overline{\gc}(1+ 2\gd ) = q'$.
Set $h = D_2g \circ \overline{\gc}$. Since $D_2g = 0$ on $A$, it 
follows that $h=0$ at 4 points of $[-2\gd , 1 + 2\gd]$. Since 
$h \in \gL _{s-1}$, Lemma~\ref{lemma-main2} implies that

\begin{eqnarray*}
| h(c) - h(0) | & \leq & \| h \| _{s-1}(1 + 4\gd )^{s-1} \\
& \leq & \| f\| _s |Q|^{s-1} (1 + 4\gd )^{s-1} . 
\end{eqnarray*}

\nin Since $\gd >0$ is arbitrary, this completes the proof.
\q

To prove a similar decomposition result for functions of class
$\gL _s$, we require the following observation of Norton
\cite{norton}

\begin{lemma}{(Norton)}
\label{lemma-main5}
If $g \in \gL _1(\reals , \reals)$, $a<b$, and $g(a) = g(b) = 0$,
then for all $x \in [a,b]$,
\[ |g(x)| \leq \| g \| _1 |b-a|. \]
\end{lemma}
\q

\section{Proof of the main theorems}

Fix $t,s >1, r \in \naturals$. To prove Theorem~\ref{thm-main1},
we consider a Banach space $E$, a normed vector space $F$, and 
a $t$-finite set $A \subset E$ of rank-$r$ points for a $C^1$
map $f\! : E \ra F$. Since $Df$ is continuous, the function 
$x\mapsto \rank (Df_x)$ is lower semi-continuous on $E$, and so
its level sets are Borel sets. Consequently, there exists a Borel
set $A'$ of rank-$r$ points of $f$ which contains $A$ and satisfies
$\calh ^t(A') = \calh (A)$.

For some $\gm , \gh >0$, we suppose for the moment that there is 
a decomposition $\{ A_i \}$ of $A'$ such that for each $i\in 
\integers _+$ and $\gd \in (0,M_i^{-1})$, choose a covering 
$\{ S_j \}$ of $A_i$ such that $|S_j| = \gel _j < \gd$ for each
$j$ and 
\[ \sum_j \gel _j^t \leq \calh ^t(A_i) + \gd .\]
For each $j$, let $S_j'$, $S_j''$ denote the orthogonal projections
of $S_j$ in the subspaces $E_i'$ and $E_i''$, respectively. Evidently,
$|S_j'| \leq |S_j|$, and so there exists an $r$-cube $Q'_j \subset
E'$ of diameter $\leq (1+r)^{1/2} \gel _j$ which contains $S_j'$.
The product $Q_j' \times S_j''$ then contains $S_j$ and has diameter
$\leq (1+r)^{1/2}\gel _j$. Given $\gm >0$, we partition $Q_j'$ into
$([\gm ^{-1} \gel _j^{1-s}] +1)^r$ equal subcubes of edge $\leq \gm
\gel _j^s$. For every such subcube $\tilde{Q}$, we have $|\tilde{Q}|
\leq r^{1/2} \gel _j$, so that by $(*)$,
\[ |f(A_i \cap (\tilde{Q} \times S_j''))| \leq (P(\gh ,\gm ) + M_i \gd )
\gel _j^s. \]
For $d = r + (t-r)/s$ and $P = P(\gh ,\gm )$, we then have
\[ \calh ^d(f(A_i)) \leq \sum_j ([\gm ^{-1} \gel _j^{1-s}] +1)^r 
((P+M_i\gd )\gel _j^s)^d \leq (P + M_i\gd )^d \gm ^{-r}
(\calh ^t(A_i) + \gd ), \]
so that tending $\gd$ to 0 gives $\calh ^d(f(A_i)) \leq P^d\gm ^{-r}
\calh ^t(A_i)$. Since the $A_i$ are disjoint Borel sets, we have
$$ \calh ^d(f(A)) \leq \sum_{i=1}^{\infty} \calh ^d(f(A_i)) \leq
P^d \gm ^{-r} \calh ^t(A).  \eqno{(**)} $$

\vspace{.1in}

\nin {\bf Proof of Theorem~\ref{thm-main1}.} To prove statement (1)
of Theorem~\ref{thm-main1}, we note that by Lemma~\ref{lemma-main}(1),
the inequality $(*)$ above holds for all $\gh, \gm >0$ with
\[ P(\gh , \gm ) = (\| f \| _1 + \gh + 1)\gm . \]
Since $d>r$, tending $\gh , \gm$ to 0 in the inequality $(**)$ gives
$\calh ^d(f(A)) =0$.

Similarly, Lemma~\ref{lemma-main}(2) implies that when $f\in \gL _s$
for $s>2$, then $(*)$ holds for 
\[ P(\gh , \gm ) = (\| f \| _1 + \gh )\gm + \| f\| _s + \gh ). \]
Thus, we obtain statement (2) of Theorem~\ref{thm-main1} by setting
$\gh = 1$ and tending $\gm$ to zero in the equality $(**)$.

Finally Lemma~\ref{lemma-main}(3) asserts if $A$ is $t$-rectifiable,
then $A\setminus A_0$ admits a decomposition of the form above for
all $\gh, \gm >0$ and 
\[ P(\gh , \gm ) = (\| f\| _1 +\gh +1)\gm . \]
Since $A_0$ is $t$-null, Theorem~\ref{thm-main1}(1) implies that
$f(A_0)$ is $d$-null, so that the inequality $(**)$ is valid for
all $\gh , \gm >0$. Since $d>r$, we conclude that $\calh ^d(f(A)) =0$.
\q

\vspace{.1in}

\nin {\bf Proof of Theorem~\ref{thm-main2}.} To prove statement
(1) of Theorem~\ref{thm-main2}, we note that by Lemma~\ref{lemma-main}(1),
the inequality $(*)$ above holds for all $\gh , \gm >0$ with
\[ P(\gh , \gm ) = (\| f \| _1 + \gh +1 )\gm. \]
Since $d>r$, tending $\gh ,\gm$ to 0 in the inequality $(**)$ gives
$\calh ^d (f(A)) = 0$.

Similarly, Lemma~\ref{lemma-main}(2) implies $(*)$ holds for
\[ P(\gh , \gm ) = (\| f \| _1 + \gh )\gm + \| f\| _s + \gh ). \]
Thus, we obtain statement (2) of Theorem~\ref{thm-main1} by setting
$\gh = 1$ and tending $\gm$ to zero in the equality $(**)$.

Finally Lemma~\ref{lemma-main}(3) asserts if $A$ is $t$-rectifiable,
then $A\setminus A_0$ admits a decomposition of the form above for
all $\gh, \gm >0$ and 
\[ P(\gh , \gm ) = (\| f\| _1 +\gh +1)\gm . \]
Since $A_0$ is $t$-null, Theorem~\ref{thm-main1}(1) implies that
$f(A_0)$ is $d$-null, so that the inequality $(**)$ is valid for
all $\gh , \gm >0$. Since $d>r$, we conclude that $\calh ^d(f(A)) =0$.
\q

\vspace{.1in}

\begin{lemma}
Let $F$ be any Banach space. Suppose $s>1$ and $a<b<c<d$ are real 
numbers, and suppose that each of the intervals $[a,b]$, $[b,c]$,
and $[c,d]$ contains at least $[s]+1$ zeros of a function
$f\in \gL _s(\reals ,F)$. Then
\[ \max |f|_{[b,c]}| \leq \| f\| _s \cdot (d-a)^s. \]
\end{lemma}

\pff If $F=\reals$, then by repeated application of the mean-value
theorem, our hypotheses imply that the derivatives $f^{(1)}, \cdots ,
f^{([s])}$ of $f$ all have zeros in $[a,b]$, $[b,c]$, and $[c,d]$.
Thus, for $k=1, \cdots ,[s]-1$,
\[ \max_{[b,c]} |f^{(k)}| \leq \max_{[b,c]} |f^{(k+1)}| (c-b). \]
Also,
\[ \max_{[b,c]} f - \min_{[b,c]} f \leq \max_{[b,c]} |f^{(1)}| (c-b). \]
Hence,
\[ \max_{[b,c]} f - \min_{[b,c]} f \leq \max_{[b,c]} 
|f^{([s])}| (c-b)^{[s]}. \]

For noninteger $s$, the proof is finished by noting that 
\[ \max_{[b,c]} |f^{[s]}| \leq \| f\| _s (c-b)^{(s-[s])}, \]
which follows from the $(s-[s])$-H\"older condition satisfied
by $f^{[s]}$.

When $s$ is an integer, it suffices to establish that
\[ \max_{[b,c]} |f^{[s]}| \leq \| f\| _s (d-a). \]
This is implied by Lemma~\ref{lemma-main5}.

For an arbitrary Banach space $F$, we can apply the preceding
argument to the function $u^* \circ f$, where $u^* \in F^*$ is
any unit covector. Since $\| u^* \circ f \| _s = \| f \| _s$,
the lemma follows.
\q

\section{Sharpness of the Theorems}

Recall that a {\em modulus of continuity} $\frakm$ is a class of monotone, continuous functions 
$\gh \! : [0,1] \ra [0,\infty )$ which are subadditive, i.e.,
\[ \gh (s+t) \leq \gh (s) + \gh (t) \]
for all $s,t$ in some subinterval $[0,a)$ of $[0,\infty )$. Any two members $\gh _1, \gh _2$ of $\frakm$ are
equivalent in the sense that 
\[ C^{-1} \leq \frac{\gh _1(t)}{\gh _2(t)} \leq C \]
for some constant $C$ and all $t$ in some subinterval $[0,a)$ of $[0,1]$. A partial order $\zzz$ is defined
for moduli of continuity by setting $\frakm _1 \zzz \frakm2$ whenever
\[ \lim_{t \ra 0^+} \frac{\gh _1 (t)}{\gh _2(t)} = \infty \]
for any $\gh _i \in \frakm _i$, $i= 1,2$.

Fix $n,m \in \integers_+$ and let $X,Y$ be Banach spaces. For any nonnegative integer $k$ and modulus of
continuity $\gh$, we define the subset $C^{k,\frakm}$ of $C^k(X,Y)$ to consist of all $f$ such that for 
each $x\in X$, there exists $\gel _x >0$ and $\gh \in \frakm$ such that 
\[ \| D^k f(y) - D^k f(z) \| \leq \gh (\| y - z\| ) \]
for all $y,z \in B(x,\gel )$. Note that if $\frakm _1 \zzz \frakm _2$, then $C^{k,\frakm _2} \subset C^{k,\frakm _1}$.

\begin{ex} For fixed $\ga \in (0,1]$, the modulus of continuity $\frakm _{\ga}$ containing the function $\gh _{\ga} (t) = t^{\ga}$
gives rise to the familiar spaces $C^{k,\frakm _{\ga}}$ of $C^k$ functions whose $k$-th derivatives satisfy
a local $\ga$-H\"older estimate.

Since $(-\log (t))^s t^{\gel} \ra 0$ as $t \ra 0$ for any fixed $\gel , s >0$, the modulus of continuity $\frakm$
containing the function $\gh (t) = (-\log (t))^s t^{\ga}$ satisfies $\frakm _{\gb} \zzz \frakm \zzz \frakm _{\ga}$ for
all $\gb < \ga$.

\trg

\end{ex}

\begin{prop}
\label{prop-sharp}

Fix $\ga \in (0,1]$. If $\frakm$ is a modulus of continuity such that $\frakm \zzz \frakm _{\ga}$, then for any
$k\in \integers _+$ and $t\in (0,1)$, there exists a $C^{k,\frakm}$ function $f\! : \reals \ra \reals$ and a 
$t$-null set $A$ of critical points of $f$ such that $f(A)$ is not $(t/(k+\ga ))$-sigmafinite.

\end{prop}

\nin To prove the proposition, we will use the following extension lemma. Given a compact subset $K \subset \reals$,
we refer to bounded components of $\reals \setminus K$ as {\em gaps} of $K$.

\begin{lemma}
\label{lemma-sharp}

Fix $k \in \integers_+$ and $\ga \in (0,1]$, and let $\gh \! : [0,1] \ra \reals$ represent a modulus of continuity
$\frakm$ such that $\frakm \zzz \frakm _{\ga}$. If $B\subset \reals$ is a compact set and $f \! : B \ra \reals$ is a 
monotone embedding such that 
\[ |g(J)| \leq |J|^k \gh (|J|) \]
for each gap $J$ of $B$, then there exists a $C^{k,\frakm }$ extension $f \! : \reals \ra \reals$ for which $B$ is a
critical set.

\end{lemma}

\pff Let $[a,b]$ denote the convex hull of $B$ and let $J_i = [a_i,b_i]$ denote its gaps. Set $l_i = |J_i|$ and 
define $t_i = |g(b_i) - g(a_i)|$.

Let $s\! : [0,1] \ra \reals$ be any smooth function such that 

\ben

\item $s>0$ on $(1/3,2/3)$ and $s=0$ otherwise, and
\item $\int _0^1 s(x) \, dx = 1.$

\een

Denote by $M_n$ the $C^n$ norm of $s$. We construct the derivative $r$ of $f$ by defining
\[  r(x) = \left\{ \begin{array}{ll}
	0            & \mbox{for $x \in \reals \setminus \cup_i J_i$} \\
        \frac{t_i}{l_i} s \left( \frac{x-a_i}{l_i} \right) 
		     & \mbox{for $x \in J_i$} \\
	\end{array}
\right.   \]
For each $x \in B$, we again let $\cali _x$ denote the set of gaps of $B$ lying to the left of $x$. By the
definitions above, we have
\[ g(x) = g(a) + \sum_{I\in \cali_x} t_i. \]
Define
\[ f(x) = g(a) + \int_a^x r(t) \, dt. \]
Then
\[ f(x) = g(a) + \sum_{I\in \cali_x} t_i \int_0^1 s(t) \, dt = g(a) + \sum_{I\in \cali_x} t_i = g(x). \]
Thus, the function $f$ is a monotone extension of the homeomorphism $g$.

Evidently $f$ is $\cinf$ on the complement of $B$ and near isolated points of $B$. If $x \in B$ is not an
isolated point, then for any $k$ there exists a neighborhood $U$ of $x$ such that $y\in U\setminus B$ implies
that $y$ lies either in a gap with endpoint $x$, or in a gap of length $\ell$ less than $|U|$. In the latter
case, property (1) of the function $s$ implies that either $f^{(n)}(y) = 0$ or $|x-y| \geq \ell /3$, and so
\[ |f^{(n)} (y)| = |r^{(n-1)}(y)| \leq M_n \ell ^{k-n} \cdot \gh (\ell ) \ell 3^{k+1}|y-x|^{k-n} \gh (|y-x|). \]
Thus, the derivative $f^{(n)}$ tends to zero on approach to $A$ for each $n \leq k$. We conclude that $f$ is
$C^{k,\frakm}$ from the preceding inequality and the following fact, whose proof is left to the reader.

\begin{lemma}
Let $K\subset \reals$ be compact and $f\! : \reals \ra \reals$ be continuous on $\reals$ and $C^k$ on 
$\reals \setminus K$. Suppose that
\ben

\item for $i = 1, \ldots , k$, $|f^{(i)}(x)| \ra 0$ uniformly as $x \ra K$, and
\item $|f(x) - f(y)| \leq o(|x-y|^k)$ uniformly for $x,y \in K$.

\een
Then $f$ is $C^k$.
\end{lemma}

Finally we note that by construction, each connected component of the critical set of $f$ intersects $B$, and
therefore the critical value set of $f$ equals $A$.
\q

\nin {\bf Proof of Proposition~\ref{prop-sharp}.} Given a sequence of positive numbers $\ell _n$ such that
\[ \sum_{n=0}^{\infty} 2^n \ell _n \leq 1, \]
we let $B(\fraks )$ denote the Cantor set obtained by an analog of the ``middle thirds'' construction in which 
$2^n$ intervals of length $\ell _n$ are removed at the $n$-th stage.

Fix $t \in (0,1)$ and define sequences by
\[ a_n = 2^{-n/t} \hspace{.25in} b_n = \gh (2^{-n/t})^{-1/2}2^{-n/t} \hspace{.25in} c_n = \gh(2^{-n/kt})^{-1/2}2^{-n/kt}
\hspace{.25in} d_n = 2^{-n/kt}, \]
and let $A,B,C,D$ denote the corresponding central Cantor sets. By Lemma~\ref{lemma-sharp}, the obvious homeomorphism
$B \ra C$ extends to a $C^{k,\frakm}$ map $f\! : \reals \ra \reals$ containing $B$ within its set of critical set.

We claim that $B$ is $t$-null, whereas $C = f(B)$ is not $(t/(k+\ga ))$-sigmafinite. Indeed, this can be checked 
either by a direct computation, or by applying Lemma~\ref{lemma-sharp} to find $C^1$ maps $g,h \! : \reals \ra \reals$
such that $g,h$ are critical on $A,C$, respectively, and $g(A) = B, h(C) = D$. The claim then follows by an 
application of Theorem~\ref{thm-main1} to the mappings $g,h$.
\q

\section{Morse Criticality in Infinite Dimensions}

For $1 \leq p < \infty$, we denote by $\ell _p$ the space of real sequences $x = (x_j)$ equipped with the 
norm
\[ \| x \| _p = \left( \sum_j |x_j|^p \right) ^{1/p}. \]
By $\ell _{\infty}$ (resp.\ $c_0$), we denote the space of bounded sequences (resp.\ sequences tending to 0) with the
usual sup-norm. For $E = \ell _p$ (or $E = c_0$), we will say that a sequence of functions $f_j \! : \reals \ra \reals$
satisfies condition $(*)_E$ if $(f_j(0)) \in \ell _p$ (or $(f_j(0)) \in c_0$), and the $f_j$ have locally uniformly
bounded derivatives, i.e., for each $k\in \naturals$, $r>0$, there exists a real constant $M_{k,r}$ such that 
\[ | f_j^{(k)}(x) | \leq M_{k,r} \]
whenever $|x| < r$.

\begin{lemma}
\label{lemma-infinite}
Fix $1\leq p < \infty$ and consider a sequence of $\cinf$ functions $f_j \! : \reals \ra \reals$ satisfying condition
$(*)_{\ell _p}$ (or $(*)_{c_0}$). Then the map $f\! : \ell _p \ra \ell _p$ (or $f \! : c_0 \ra c_0$) given by $f(x)
= (f_j(x_j))$ is $\cinf$ Fr\'echet differentiable.
\end{lemma}

\pff From condition $(*)_{\ell _p}$ and the mean-value theorem, it follows that 
\[ \| f(x) - f(y) \| _p \leq M_{1, \| x\|_p + \| y\|_p } , \]
i.e., $f$ is a well-defined, locally Lipschitz endomorphism of $\ell _p$. For each $k\in \naturals$, define 
$f^{\{ k \}} \! : \ell _p \ra \ell _{\infty}$ by
\[ f^{\{ k \} } (x) = (f_j^{\{ k \}}(x_j)) . \]
A simple computation shows that each $f^{\{ k \} }$ is differentiable, and )
\[ Df^{\{ k \} }(x)(y) = (f_j^{\{ k \} }(x)(y) = (f_j^{\{ k \}}(x_j) y_j). \]
The linear map $T\! : \ell _{\infty} \ra L(\ell _p,\ell _p)$ given by $T(x)(y) = (x_iy_i)$ satisfies
\[ \| T(x)(y) \|_p = \left( \sum_j |x_jy_j|^p \right) ^{1/p} \leq \| x \| _{\infty} \| y \|_p \]
and is therefore continuous. By the preceding remarks, we have
\[ Df^{\{ k \}} = T \circ f^{\{ k+1 \}}. \]
>From this equation, a simple inductive argument shows that each $f^{\{ k \}}$ is $\cinf$ Fr\'echet differentiable.
Specifically, each $f^{\{ k \}}$ is continuous. Assuming that each $f^{\{ k \}}$ is $C^n$, the preceding equation
implies that $Df^{\{ k \}}$ equals the composition of $f^{\{ k+1 \}}$ with the bounded linear map $T$, and 
so $f^{\{ k \}}$ is $C^{k+1}$.
\q

By introducing points of infinite singularity, we can improve the target space of $f$ as follows.

\begin{lemma}
Fix $1\leq p < \infty$ and consider a sequence of $\cinf$ functions $f_j \! : \reals \ra \reals$ with locally uniformly
bounded derivatives and $f_j^{\{ k \}}(0) = 0$ for all $k \in \naturals, j\in \integers_+$. Then the map $f\! : \ell _p
\ra \ell _1$ given by $f(x) = (f_j(x_j))$ is $\cinf$ Fr\'echet differentiable.
\end{lemma}

\pff For each $j\in \integers _+$, $k\in \naturals$, our hypotheses combined with Taylor's theorem imply that 
\[ | f_j (x_j) - f_j(0) | \leq M_{[p]+1, |x_j|} |x_j|^p, \]
and so
\[ \| f(x) \|_1  = \| f(x) - f(0) \|_1 \leq M_{[p]+1,\| x\|_p} \| x\|_p^p \]
for all $x \in \ell _p$. Thus, $f$ is a well-defined map from $\ell _p$ into $\ell _1$. A similar argument shows that 
for each $k\in \naturals$, then map $f^{\{ k \}}$ defined in 
Lemma~\ref{lemma-infinite} is also a well-defined map 
from $\ell _p$ into $\ell _1$. Moreover, a simple computation again shows that 
\[ Df^{\{ k \}} (x)(y) = (f_j^{\{ k+1 \}}(x_j)y_j). \]
Evidently the linear map $T\! : \ell _1 \ra L(\ell _p, \ell _1)$ given by $T(x)(y)$ satisfies
\[ \| T(x)(y) \| _1 \leq \| x \|_1 \| y \| _p \]
and is therefore continuous. As in Lemma~\ref{lemma-infinite}, we have $Df^{\{ k \}} = T\circ f^{\{ k+1 \}}$, and 
the assertion follows by the same argument as before.
\q

\begin{ex}
Consider the polynomial map $f\! : \ell _1 \ra \ell _1$ given by the component functions
\[ f_j(x) = 3\cdot 2^{-j\ga} x^2 - 2x^3. \]
A check of the definitions shows that the critical set of $f$ contains the Cantor set
\[ \calc = \{ (\gb _j) : \gb _j \in \{ 0, 2^{-j\ga} \} \} \]
of rank-0 points for $f$, and 
\[ f(\calc ) = \{ (\gb _j) : \gb _j \in \{ 0, 2^{-3j\ga} \} \} . \]
The Hausdorff dimensions of $\calc$ and $f(\calc )$ are $\ga ^{-1}$ and $(3\ga )^{-1}$, respectively, and both
have positive, finite measure in these dimensions.

Denoting by $s \! : \ell _1 \ra \reals$ the continuous linear map 
\[ s(x) = \sum_j x_j, \]
we note that $\calc$ is also contained in the critical set of the composition $g = s\circ f$. Thus, the critical
value set of $g$ contains
\[ g(\calc ) = \left\{ \sum_j \gb _j : \gb _j \in \{ 0, 2^{-3j\ga } \} \right\} , \]
which has positive, finite Hausdorff measure in dimension $(3\ga )^{-1}$. Finally, we note that in the special
case of $\ga = 1/3$, the set $g(\calc )$ equals the unit interval $[0,1]$.
\end{ex}

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