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\begin{document}
\begin{center}
{\Large {\bf Zeros of polynomials with 0,1 coefficients}} \\
\vspace{\baselineskip}
{\em A. M. Odlyzko} \\
{\em B. Poonen}* \\
\vspace{0.5\baselineskip}
AT\&T Bell Laboratories \\
Murray Hill, New Jersey 07974 \\
\footnotetext[1]{Current address: Dept. of Mathematics, Univ. of
California, Berkeley, California 94720.}
\vspace{1.5\baselineskip}
{\bf ABSTRACT}
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\end{center}
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\hspace*{.25in}
Zeros of polynomials with $0,1$ coefficients exhibit many interesting
features,
including fractal appearance.
This paper
obtains bounds for such zeros.
It shows
that zeros with a
sufficiently large negative
real part are real.
It also proves that the closure of the
set of these zeros is path connected.
\clearpage
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\begin{center}
{\Large {\bf Zeros of polynomials with 0,1 coefficients}} \\
\vspace{\baselineskip}
{\em A. M. Odlyzko} \\
{\em B. Poonen}* \\
\vspace{0.5\baselineskip}
AT\&T Bell Laboratories \\
Murray Hill, New Jersey 07974 \\
\footnotetext[1]{Current address: Dept. of Mathematics, Univ. of
California, Berkeley, California 94720.}
\vspace{1.5\baselineskip}
\end{center}
\setlength{\baselineskip}{1.5\baselineskip}
\section{Introduction}
\hspace*{\parindent}
Zeros of polynomials with random coefficients occur in many
scientific and engineering problems.
A general overview of the subject and references can be found in
the book of Bharucha-Reid and Sambandham \cite{BRS}, which is
the basic reference on this topic.
There is a wealth of information about distribution of zeros in the
complex plane and on the real line.
Almost all of the results are for coefficients chosen independently
from a common distribution that is continuous, and usually Gaussian.

In this paper we consider zeros of polynomials with $0,1$
coefficients.
These zeros have some features that distinguish them from those of the commonly considered families of random polynomials.
Let
\beql{eq101}
P = \left\{ f(z):
f(z) = 1 + \sum_{j=1}^d a_j z^j ,~~
a_j = 0 ~\mbox{or} ~1 ~~\mbox{for all} ~~j \right\} ~.
\eeq
(We exclude polynomials with constant term 0,
as their zeros, other than 0, are those of polynomials of lower degree with coefficients $0,1$.)
Define
\beql{eq102}
W = \{ z \in \CC: f(z) =0 ~~\mbox{for some}~~ f \in P \} ~.
\eeq
For each degree $d$, there are $2^{d-1}$ polynomials $f(z) \in P$
of degree $d$, and so $W$ is a countable set.

There are few published results about $W$.
In \cite{BFO} it was shown that ${\rm Re}~ (z) < 3/2$ for all $z
\in W$. This was used to prove that if $f(2)$ is a prime
for some $f(z) \in P$, then $f(z)$ is irreducible over the
rationals.
(For further results relating zeros to irreducibility, see
\cite{Fil}.
It is conjectured that almost all $f(z) \in P$ are
irreducible, but this is still open.
This is in contrast to the case of fixed degree polynomials when
the range over which the coefficients are allowed to run
increases.
There it is known that almost all polynomials are not only
irreducible, but also have $S_n$ as their Galois group.
For latest results and references on this topic, see
\cite{Gal}.)

Our results are best illustrated by pictures of zeros.
Figure~1 shows all zeros of
the
polynomials with
coefficients $0,1$ of degrees $\le 16$, and with constant term 1,
except for the negative real zeros that lie are $< -1.5$.
We show that $W$ lies
between the curves
\beql{eqA103}
C_1 = \left\{
z :~ |z| \le 1 , ~ \frac{|z|}{1- |z|} =
\left| \frac{2-z}{1-z} \right| \right\}
\eeq
and
\beql{eqA104}
C_2 = \left\{ z: ~ |z| \ge 1 , ~
\frac{|z|}{|z|-1} =
\left | \frac{2-z}{1-z} \right| \right\} ~.
\eeq
The curve $C_1$ is mapped to $C_2$ by $z \to 1/z$.
This mapping takes $W$ to itself, since if $z \in W$, and $z$ is a root of $f(z) \in P$ and
$\deg f(z) = d$, then $1/z$ is a root of $z^d f(1/z) \in P$.
We show that all $z \in W$ are enclosed strictly between $C_1$
and $C_2$.
From this it follows that
for all $z \in W$,
\beql{eq103}
\frac{1}{\vp} < |z| < \vp ~,
\eeq
where
\beql{eq104}
\vp = \frac{1+ 5^{1/2}}{2}
\eeq
is the ``golden ratio.''
(The bound \eqn{eq103}
has been proved independently in different contexts by
Flatto, Lagarias, and Poonen \cite{FL} and by Solomyak
\cite{Sol}.)
We also show that the line segment $[- \vp , -\vp^{-1}] \in \overline{W}$.
However, $- \vp \not\in W$ and $- \vp^{-1} \not\in W$.
Further, there is a constant $\dt > 0$ such that if $z \in W$,
$|z| \le \vp^{-1} + \delta$, then $z \in R$.
Thus the ``spike'' along the negative real axis that is visible
in Fig.~1, connecting curves $C_1$ and $C_2$ with the exception
of a small gap at $-1$, is due to zeros.

Since polynomials in $P$ have nonnegative coefficients,
$1 \not\in W$.
However, since $\zeta \in W$ for every root of unity $\zt \neq
1$, $1 \in \overline{W}$, where $\overline{W}$ denotes the
closure of $W$.
We answer a question posed by J.~H. Conway and Richard Parker
about the behavior of $W$ near 1 by proving there exist points
$z=x+iy \in W$ such that $0 < x-1 = o(|y|)$, so that these
points come in tangent to the $x$-axis.

Figure~2 shows the zeros of polynomials $f(x) \in P$ of degree 18
that are close to $z=1$.
Figure~3 shows zeros of polynomials $f(x) \in P$ of all degrees
$\le 32$ that fall in a certain small region of the complex
plane.
Figures~4, 5, and 6 show pictures of parts of $\overline{W}$.
The region depicted in Figure~4 is the same as that of
Figure~3.
Section~6 explains how these pictures were created.

Theorem~\ref{t201} of Section~2, which says that $W$ is
contained between $C_1$ and $C_2$, is not best possible.
The only points of $\overline{W}$ that are in $C_1 \cup C_2$ are
1, $- \vp$, $- \vp^{-1}$.
In Section~6 we will show how to obtain more precise bounds for
$W$.
However, because of the fractal nature of $W$, there is no simple
description of its shape.

Many features visible in the graphs can be explained (at least
heuristically, and often rigorously) by using known results or
methods.
When one graphs zeros of any single polynomial with coefficients
$0$ and $1$, most of them are close to the unit circle $|z|=1$
and they are equidistributed in angles, so that the first
quadrant, for example, has close to $1/4$ of the total.
This phenomenon is true for all polynomials whose coefficients
do not vary much, as follows from results originating with
Erd\"{o}s and Tur\'{a}n \cite{ET}.
For statements and references to general results, see
\cite{BRS}.

The expected number of real roots of a random polynomial (which have to be negative for $f(z) \in P$) grows logarithmically with $n$,
as was first noted by Kac and Rice (see \cite{BRS}).
Furthermore, the variance is small.

In Figures~1 and 2, there is a perceptible clustering of zeros.
This is a reflection of the ``averaging phenomenon'' for roots
of random polynomials \cite{BRS,SB}, and again is not special to
$0,1$ coefficients.
The ``average'' of the polynomials of degree $n$ that are in $P$
is
\beql{eq105}
g(z) = z^{n} +1+ \frac{1}{2}
\sum_{k=1}^{n-1} z^k =
\frac{(1-2z)z^{n} -z+2}{2(1-z)} ~,
\eeq
and on average the zeros of $f(z) \in P$ tend to cluster near the zeros of
$g(z)$.

Figures~1 and 2 show several large ``holes,''
which contain either just one or no zeros.
These holes are usually centered at algebraic integers $\af$ of low degree and small height (i.e.,
algebraic integers $\af$ that satisfy
polynomial equations with small integral coefficients).
The most prominent of the holes are at the roots of unity,
such as $-1$ and $i$.
As one computes zeros of polynomials $f(z) \in P$ of increasing degrees, the
large holes in Figures~1 and 2 fill up.
However, there are other holes, such as those visible in
Figures~3--6, that are free of zeros even when the degree
increases.

We show in Section~3 that there is an open neighborhood of $\{ z: |z| =1 ,
z \neq 1 \}$ that is in $\overline{W}$.
In Section~4 we prove that
$\overline{W}$ is connected.
The more involved argument in Section~5 proves that $\overline{W}$ is path connected.
Since the unit circle is contained in $\overline{W}$, but $0 \not\in \overline{W}$, $\overline{W}$ is clearly not simply connected.
Numerical experiments suggest that $\overline{W}$ has
``holes'' in it besides the big hole containing 0.
(That is, $\CC \setminus \overline{W}$ has more than 2 connected components.)
In particular, the disk of radius $10^{-5}$ centered at $-0.69098 + 0.33062~i$ appears to be part of such a hole.
This hole and some neighboring ones are pictured in Figures~5
and 6.
Other, even larger holes, can be seen in Figures~3 and 4.

$W$ has a fractal appearance that is reminiscent of some of the Julia sets
\cite{B,D}.
In Section~6
we sketch arguments that explain how this arises.
However, we do not have estimates for such interesting parameters
as the Hausdorff dimension of the boundary of $\overline{W}$.

In contrast to our result that $\overline{W}$ is path connected,
the Mandelbrot set is only known to be
connected, although it is conjectured to be path connected \cite{B,D}.
Our methods are simpler than those used to study the connectedness of the Mandelbrot set.
They are similar to the techniques developed for investigating iterated function systems \cite{B}.

Results
similar to those for polynomials with
$0,1$ coefficients can also be obtained for other families of
polynomials with a small set of possible coefficients.
For example, for $\pm 1$ coefficients,
pictures of zeros are qualitatively similar to those of $0,1$ polynomials.
There is symmetry about the imaginary axis as well as the
real axis (corresponding to changing the variable $z$ to $-z$).
There are two ``spikes''
of zeros along the real axis that fill the intervals
$[-2, - 1/2]$ and $[1/2,2]$, while there are no other
zeros in $|z| \le 1/2 + \delta$ or $|z| \ge 2- \delta$ for some $\delta > 0$.
For polynomials with cubic roots of unity as coefficients, there are no
``spikes,''
but the zeros still have a fractal appearance.

The set
\beql{eq106}
\overline{W} \cap \{ z: |z| < 1 \}
\eeq
is the set of zeros of power series
\beql{eq107}
f(z) = 1+ \sum_{k=1}^\In a_k z^k ,~~~~a_k = 0 ~\mbox{or}~ 1 ~.
\eeq
Since $z^{-1} \in W$ for all $z \in W$, it is sufficient to study $z \in W$, $|z| \le 1$, and in
some ways it is more natural to deal with the above power series.

Some of our methods and results are similar to those of Thierry
Bousch \cite{Bo1,Bo2}, whose work was brought to our attention
by D. Zagier.
The report \cite{Bo1} proves that the closure of the set of
zeros of polynomials with coefficients 0, $\pm 1$ is connected.
The thesis \cite{Bo2} contains, along with a variety of other
results, general methods for studying similar problems.
In the area where our work overlaps \cite{Bo1,Bo2}, we obtain a
somewhat stronger result by proving path connectivity.

Boris Solomyak \cite{Sol} has studied zeros of power series of
the form \eqn{eq107}, but with the $c_k$, $k \ge 1$, allowed to
take any real values in the interval $[0,1]$.
He shows that the bound \eqn{eqA203} holds there as well, and that
there is a ``spike'' of real zeros along the negative real axis.
However, the zeros of Solomyak's functions are substantially
different from those we investigate.
For example, he shows that segments of the boundary he
investigates have everywhere dense sets of points where a
tangent exists, as well as everywhere dense sets of points with
no tangent.
There are also no holes in Solomyak's set of zeros.

The paper of Brenti, Royle, and Wagner \cite{BRW} discusses
various properties of chromatic polynomials.
While it is not directly related to our work, the numerical
evidence it presents shows that zeros of chromatic polynomials
may also exhibit fractal behavior.
This may also be true for the partition function zeros of
\cite{BJ}.
\section{Bounds and locations for zeros}
\hspace*{\parindent}
A polynomial $f(z) \in P$ can have multiple zeros.
If $\zt \neq 1$ is a $d$-th root of unity,
then $\zt$ is a zero of
$$g(z) = \sum_{j=0}^{d-1} z^j ~,$$
and therefore a zero of $g(z^k)$ for any $k$ such that $d | k-1$.
Hence it is a zero of multiplicity 2 for $g(z) g(z^k)$, a polynomial in $P$.
Higher multiplicities can be obtained by iterating this procedure.
On the other hand, we do not know whether any $z \in W$ that is not a root of unity can be a multiple root of any $f(z) \in P$.
There do exist power series with coefficients $0,1$ that have double
zeros $z$ with $|z| <1$,
as will be shown in Section~3.

Inside a disk $\{z : |z| < r \}$ for $r < 1$, any polynomial $f(z) \in P$
can have only a bounded number of zeros.
We prove a slightly more general result that will be used later
on.
\begin{prp}
\label{pr201}
Suppose that $f(z)$ is a power series of the form
$$f(z) =1 + \sum_{k=1}^\In a_k z^k ,~~~~a_k =0,1~.$$
Then for any $r$, $0 < r < 1$, $f(z)$ has
\beql{eq202}
\le 2 ( - \log (1-r^{1/2} )) ( - \log r )^{-1} ~
\eeq
zeros in $|z| \le r$.
\end{prp}

\pf
We apply Jensen's theorem (Theorem~3.61 of \cite{T}).
If $z_1 , \dd , z_n$ are the zeros in $|z| < R$, where $r < R <
1$,
then we find that
\beql{AMOeq2}
\sum_{j=1}^n \log (R/ |z_j |) =
\frac{1}{2 \pi} \int_0^{2 \pi}
\log | f(Re^{i \theta} ) | d \theta ~,
\eeq
since $f(0) =1$.
Therefore, if $m$ is the number of zeros in $|z| < r$, we have
\beql{AMOeq3}
m( \log R - \log r ) \le \frac{1}{2 \pi} \int_0^{2 \pi}
\log | f(Re^{i \theta} )|~d \theta ~.
\eeq
Since
$$| f( Re^{i \theta} ) | \le
\sum_{k=0}^\In R^k = (1-R)^{-1} ~,$$
we obtain
$$m \le (- \log (1-R)) ( \log R - \log r )^{-1} ~.$$
We now choose $R = r^{1/2}$, and this yields the bound
\eqn{eq202}.
(Better bounds can be obtained by selecting $R$ more carefully or estimating
the integral of $\log | f(z)|$ in Jensen's theorem better.)~$\blacksquare$

We next consider bounds on the size of $z \in W$.
Since $1/z \in W$ for $z \in W$, it suffices to
consider $|z| \le 1$.
\begin{theorem}
\label{t201}
Suppose that $z$ satisfies $|z| < 1$ and that $f(z) =0$ for some power series
of the form \eqn{eq107}.
Then
\beql{eqA203}
\frac{|z|}{1- |z|} \ge \left| \frac{2-z}{1-z} \right| ~,
\eeq
and $|z| \ge \vp^{-1}$, with equality if and only if
$z = - \vp^{-1}$ and $f(z) = 1+z+z^3 +z^5 + \cdots~$.
Furthermore, there exists a $\delta > 0$ such that if $|z| < \vp^{-1} + \delta$, then $z$ is a negative real number.
\end{theorem}

\pf
We note that
\beql{eq203}
\begin{array}{lll}
f(z) & = & 1 + \ds_{k=1}^\In a_k z^k ~=~ 1 + \df{1}{2}
\ds_{k=1}^\In z^k + g(z) \\
~~ \\
& = & \df{2-z}{2(1-z)} + g(z)
\end{array}
\eeq
where
\beql{eq204}
g(z) = \frac{1}{2} \sum_{k=1}^\In \ep_k z^k~,
~~~\ep_k = \pm 1~~\mbox{for all} ~~ k~.
\eeq
Now for $|z| < 1$,
\beql{eq205}
| g(z) | \le \frac{|z|}{2(1-|z|)} ~,
\eeq
and so we conclude that if
\eqn{eqA203} is violated, then $f(z) \neq 0$.
Equality is possible in
\eqn{eq205} only if $z$ is real, and we easily check that for
$z \in (-1,1)$,
$$\frac{|z|}{1-|z|} = \left| \frac{2-z}{1-z} \right|$$
only at $z = - \vp^{-1}$.
For $z = - \vp^{-1}$, equality holds in \eqn{eq205} when
$\ep_k = (-1)^{k-1}$ for $k \ge 1$; i.e., when
$$f(x) = 1 + x + x^3 + x^5 \cdots ~.$$

Let $\Delta = \{ z : |z| < \vp^{-1} \}$.
Then $z \mapsto \frac{2-z}{1-z}$ maps $\Delta$ to the interior of the circle
one of whose diameters is
$$\left[ \frac{2+ \vp^{-1}}{1+ \vp^{-1}} ,
\frac{2- \vp^{-1}}{1- \vp^{-1}} \right] =
[ \vp , 2+ \vp ]$$
but $z \mapsto \frac{|z|}{1-|z|}$ maps $\Delta$ to
$$\left[ 0 , \frac{\vp^{-1}}{1- \vp^{-1}} \right) = [0, \vp )$$
so \eqn{eqA203}
fails if $z \in \Delta$.
Moreover, if
$z \in \partial \Delta$, \eqn{eqA203} still fails unless
\begin{eqnarray*}
\frac{2-z}{1-z} & = & \vp \\
z & = & \frac{\vp -2}{\vp -1} ~=~ - \vp^{-1} ~.
\end{eqnarray*}

We next prove that the only $z \in \overline{W}$ with $|z|$ close to $\vp^{-1}$ are negative real numbers.
Since $\overline{\Delta}$ intersects the closed set
$$\left\{ z : |z| \le 0.9 ~~\mbox{and}~~ \frac{|z|}{1- |z|}
\ge \left | \frac{2-z}{1-z} \right| \right\}$$
only at $z= - \vp^{-1}$, there exist $\dt_1$, $\dt_2 \in (0, 10^{-10})$ such that
\eqn{eqA203} fails for $z$ in
\beql{eq209}
S = \{ z : |z| \le \vp^{-1} + \delta_1 , ~
| z + \vp^{-1}| \ge \delta_2 \} ~.
\eeq
It only remains to find the possible
elements of $\overline{W}$ that lie in
\beql{eq2010}
T = \{ z : |z + \vp^{-1} | < \delta_2 , ~
|z| < \vp^{-1} + \delta_1 \} ~.
\eeq
For $z \in T$,
\beql{eq2011}
{\rm Re} \lt 1 + \frac{z}{2(1-z)} \rt \ge
\vp^{-1} - 10 | z + \vp^{-1} | \ge q^{-1} - 10^{-9} ~,
\eeq
so if $z \in W$, then we must have
${\rm Re} ~g(z) \le - \vp^{-1} + 10^{-9}$.
Since $|g' (z) | \le 10$ for $z \in T$,
and $|g ( - \vp^{-1}) | \le \vp^{-1}$, to achieve
${\rm Re}~ g(z) \le - \vp^{-1} + 10^{-9}$,
we must have $\ep_k = (-1)^{k-1}$ for $1 \le k \le 20$, say.
Then
\beql{eq2012}
f(z) = 1+ z + z^3 + \cdots + z^{19} + h(z) ~,
\eeq
where
\beql{eq2013}
h(z) = \sum_{k=21}^\In a_k z^k ~, ~~~a_k = 0 ~~\mbox{or}~~1 ~.
\eeq
Hence for $|z| =r$,
$$
| h(z) | \le \frac{r^{21}}{1-r} ~,$$
while
\beql{eq2014}
1+z+z^3 + \cdots + z^{19} =
1+ z \frac{1-z^{20}}{1-z^2} =
\frac{1+z-z^2 -z^{20}}{1-z^2} ~.
\eeq
On the circle $|z| =r = 0.7$,
$$
| 1+z-z^2 | =
|z-\vp | | z+ \vp^{-1} | \ge 0.08 \cdot 0.9 \ge 0.07 ~,$$
so
\beql{eq2015}
\left| \frac{1+z-z^2 -z^{20}}{1-z^2} \right| \ge
\frac{0.07 - (0.7)^{20}}{1+(0.7)^2} \ge 0.03 ~.
\eeq
On the other hand, $(0.7)^{21} / 0.3 < 0.01$, so by Rouch\'{e}'s theorem $f(z)$ and
$(1+z-z^2 -z^{20} ) / (1-z^2)$ have the same number of zeros inside
$|z| \le 0.7$.
By the earlier part of the argument, and another application of Rouch\'{e}'s
theorem, $1+z - z^2$ and $1+z-z^2 -z^{20}$ have the same number of
zeros inside $|z| \le 0.7$, namely one.
Therefore $f(z)$ has exactly one zero inside $|z| \le 0.7$, and since $f(z)$ has real coefficients, this zero has to be real. \hfill $\blacksquare$


The argument presented above is inefficient, and shows only that some value of $\delta < 10^{-10}$ is allowable.
With a little more care one could show by an extension of the
method used above that $\delta = 0.7 - \vp^{-1} = 0.081 \dd$ is allowable,
so that any $z \in W$ with $|z| < 0.7$ is real.
In Section~6 we present a variation of this method that uses
machine computation instead of careful estimates to establish rigorously that
$\delta = 0.7 - \vp^{-1}$ is allowable.
Numerical evidence suggests that the minimal value of $|z|$ over $z \in \overline{W} \setminus \RR$ is about 0.734.
The method of Section~6 can be used to obtain estimates for the
minimal value of $|z|$ over $z \in \overline{W} \setminus \RR$
that are as accurate as one desires.

By Proposition~\ref{pr1} of the next section,
$(-1, - \vp^{-1} ] \subseteq \overline{W}$.
Since $\overline{W}$ is stable under $z \mapsto 1/z$ and closed,
it follows that
$[- \vp , - \vp^{-1} ] \subseteq \overline{W}$.

In \cite{BFO} it was shown that $z \in W$ implies $\mbox{Re} (z) < 3/2$.
Theorem~\ref{t201} immediately leads to the bound ${\rm Re} (z) < 1.22$
for $z \in W$.
Numerical evidence suggests that ${\rm Re} (z) < 1.14$ for $z \in W$.
There are $z \in W$ with ${\rm Re} (z) > 1.13$.
The methods outlined in Section~6 can be used to obtain
precise bounds.

We can analyze inequality \eqn{eqA203} for $z$ close to 1.
We find that for $z =1-x +iy$ with $x$ and $y$ small,
$x > 0$, if $|y| \le x^{3/2}$ then \eqn{eqA203}
fails,
so $z \not\in W$.
We next show that there are points in $W$ which approach 1 along trajectories tangent to the real axis.
\begin{prp}
\label{pr202}
There exists a sequence of points $z_n \in W$ such that $z_n \to
1$ as $n \to \In$ and
\beql{AMOeq16}
| {\rm Im} \, (z_n -1 ) | = o ( {\rm Re} \, (z_n -1))
~~~\mbox{as}~~~n \to \In~.
\eeq
\end{prp}

\pf
Consider the polynomial
\beql{eq2030}
f_{m,n} (z) = 1+z+ \cdots + z^{m-1} +z^n
\eeq
with $m \le n$.
For $n$ large compared to $m$, we will show that $f_{m,n} (z)$ has a zero near to
\beql{eq2031}
\af = \af_{m,n} = \exp ( \pi i /n + ( \log m ) /n ) ~,
\eeq
and ${\rm Re}\,( \af -1) \sim ( \log m ) {\rm Im} \, ( \af )$.
We show that one can take $m \lesssim n / ( \log n )$.

To show that $f_{m,n} (z)$ has a zero $\beta$ near $\af = \af_{m,n}$, let
\beql{eq2032}
g(z) = m+z^n ~.
\eeq
Then $g( \af ) =0$.
Consider
the circle
$|z- \af | = (10n)^{-1}$.
On this circle,
$| g(z) | \ge m/100$, while
\beql{eq2033}
| ( 1+z+ \cdots + z^{m-1} ) - m |
\le \sum_{k=1}^{m-1} |z^k -1 | = O (m^2 / n) ~,
\eeq
so for $m=o(n)$, by Rouch\'{e}'s theorem $g(z)$ and $f_{m,n} (z)$ have the same
number of zeros inside the circle, namely one.
This proves the claim
and answers the Conway-Parker question. \hfill $\blacksquare$
\section{A neighborhood of the unit circle}
\hspace*{\parindent}
In this section we prove that an open neighborhood of $\{ z : |z| =1 ,~z \neq 1 \}$ is contained in $\overline{W}$.
\begin{lem}
\label{le1}
If $B \subseteq \CC$ is compact, $n \ge 1$, $|z| < 1$, and
\beql{eq301}
B \subseteq \bigcup_{\ep_1 , \ep_2 , \dd , \ep_n \in \{0,1\}}
\left[ \lt \sum_{i=1}^n \ep_i z^i \rt + z^n B \right]
\eeq
then every element of $B$ is expressible in the form
\beql{eq302}
\sum_{i=1}^\In \ep_i z^i ~, ~~~~ \ep_i \in \{0,1\} ~.
\eeq
In particular,
if $-1 \in B$, then $z \in \overline{W}$.
\end{lem}

\pf
Given $b_m \in B$, inductively pick $b_{m+1} \in B$ and $\ep_{mi} \in \{0,1\}$,
$m \ge 0$, $1 \le i \le n$ such that
$$b_m = \lt \sum_{i=1}^n \ep_{mi} z^i \rt + z^n b_{m+1}~.$$
Successive substitution yields
$$b_0 = \lt \sum_{m=0}^{M-1} \sum_{i=1}^n \ep_{mi} z^{mn+i} \rt
+ z^{Mn} b_M ~.$$
Since $B$ is compact, $z^{Mn} b_M \to 0$
as $M \to \In$, so
$$b_0 = \sum_{m=0}^\In \sum_{i=1}^n \ep_{mi} z^{mn+i}~,$$
which is the desired form. \hfill $\blacksquare$
\begin{prp}
\label{pr1}
If $z \in \RR$, $-1 < z \le - \vp^{-1}$,
then $z \in \overline{W}$.
\end{prp}

\pf
Let $B = [-1 , -z ]$.
Then,
since $-1 < z \le - \vp^{-1}$ implies $z-z^2 \le -1$, we have
\begin{eqnarray*}
(z+zB) \cup zB & = &
[z-z^2 , 0] \cup [-z^2 , -z] \\
& = & [z-z^2 , -z] \\
& \supseteq & [-1 , -z] \\
& = & B ~.
\end{eqnarray*}
We now apply
Lemma~\ref{le1} with $n=1$, and conclude that $z \in \overline{W}$. \hfill $\blacksquare$
\begin{lem}
\label{le2}
If $B \subseteq \CC$ is compact,
$-1 \in B$, $n \ge 1$,
$x \in \CC$ and
\beql{eq303}
B \subseteq  {\rm int} \bigcup_{\ep_1 , \dd , \ep_n \in \{0,1\}}
\left[ \lt \sum_{i=1}^n \ep_i x^i \rt + x^n B \right]~,
\eeq
where {\rm int}$~S$ denotes the interior of $S$,
then there is a neighborhood $N$ of $x$ such that
$$N \cap \{ z : |z| < 1 \} \subseteq \overline{W} ~.$$
\end{lem}

\pf
Condition~\eqn{eq303}
implies that \eqn{eq301}
holds for $z$ in a neighborhood
of $x$, so Lemma~\ref{le2} follows from Lemma~\ref{le1}. \hfill $\blacksquare$
\begin{lem}
\label{le3}
If $B = \{ z: |z| \le R \}$ for some $R \ge 1$,
$n \ge 1$, $|x| =1$ and
\beql{eq304}
B \subseteq {\rm int} \bigcup_{j=1}^n (x^j + B)~,\eeq
then
$$x \in {\rm int} \, \overline{W} ~.$$
\end{lem}

\pf
Since $\overline{W}$ contains the unit circle and is closed
under $z \mapsto \frac{1}{z}$, this follows trivially from Lemma~\ref{le2}. \hfill $\blacksquare$
\begin{prp}
\label{pr2}
If $|x| =1$, $x \neq \pm 1$, then $x \in {\rm int} \, \overline{W}$.
\end{prp}

\pf
We claim that if $R \ge 2$, then the condition \eqn{eq304} of Lemma~\ref{le3}
holds for $n$ large enough.
If $x = \exp ( \pi i \af )$ and $\af$ is irrational, then by Kronecker's theorem
$\{ x^j : j \ge 1 \}$ is dense on the unit circle,
and then for every $\delta > 0$, the disk of radius $R+1 - \delta$ is contained in the union on the right side of \eqn{eq304} for $n$ large enough.
If $\af$ is rational, then the $\{ x^j : j \ge 1 \}$ are the vertices of a regular $k$-gon, and $k \ge 3$ since $x \neq \pm 1$.
In that case the union on the right side of \eqn{eq304}
contains a disk of radius $r$, where $r$, 1, and $R$ are the sides of a triangle, and the angle between the sides of lengths $r$ and $1$ is $\pi / k$.
Therefore, by the Law of Cosines,
$$R^2 = 1 + r^2 - 2r \cos ( \pi /k ) ~,$$
and so
$$r = \cos ( \pi / k ) + ( \cos^2 ( \pi / k ) + R^2 -1)^{1/2} ~.$$
Since $\cos ( \pi /k ) \ge \cos ( \pi /3 )$, we find that
$$r \ge 1/4 + (R^2 - 3/4)^{1/2} \ge R+ 1/20$$
for $R \ge 2$, since
$(R^2 - 3/4)^{1/2} -R$ is an increasing function of $R$. \hfill $\blacksquare$

Proving $-1 \in {\rm int} \, \overline{W}$ is trickier,
because it will not do to take $B$ as a disc of radius $\ge 1$ if
${\rm Im} \, (z)$ is small compared to ${\rm Re} \, (z+1)$.
We will instead take $B$ as a parallelogram that becomes flatter and flatter as
${\rm Im} \, z \to 0$.
The following two lemmas will be used in verifying the condition of Lemma~\ref{le1}.

\begin{lem}
\label{le4}
Let $\mbox{\tiny $\left( \begin{array}{rr} -1 & 0 \\ 1 & -1 \end{array} \right)$}$.
Let $v_j = T^j \mbox{\tiny $\lt \begin{array}{c} 1 \\ 0 \end{array} \rt$}
= (-1)^j {\binom{1}{-j}}$.
Then for $n \ge 16$
$$
\left\{ \sum_{j=1}^n \ep_j v_j :~ \ep_j \in \{0,1\} \right\}$$
contains $\left \{ {\binom{a}{b}} : a,b \in \ZZ , |a| \le 1, |b| \le n-16 \right\}$.
\end{lem}

\pf
Given such ${\binom{a}{b}}$,
first pick $\ep_1$, $\ep_2$ so that $\ep_1 v_1 + \ep_2 v_2$ has first coordinate $a$.
Next pick $\ep_3 = \ep_4 =0$ or 1 so that
$\ep_1 v_1 + \ep_2 v_2 + \ep_3 v_3 + \ep_4 v_4$ has first coordinate $a$
and second coordinate $b'$ with $b' \not\equiv b~\bmod~2$.
Certainly $|b'| \le 1+2+3+4 = 10$, so $|b-b'| \le n-6$.
If $b > b'$, then
\begin{eqnarray*}
\lefteqn{\ep_1 v_1 + \ep_2 v_2 + \ep_3 v_3 + \ep_4 v_4 +v_5 + v_{5+b-b'}} \\
&& = (a,b') - (1,-5) + (1,5+ b-b') \\
&& = (a,b) ~.
\end{eqnarray*}
If $b < b'$, then
\begin{flushright}
$
\begin{array}{l@{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}l}
\ep_1 v_1 + \ep_2 v_2 + \ep_3 v_3 + \ep_4 v_4 +v_6 + v_{6+b'-b} \\
~~ \\ [-.09in]
~~~~~~= (a,b') + (1,-6) - (1,6 +b' -b) \\
~~ \\ [-.09in]
~~~~~~= (a,b) ~.~~~ & \blacksquare
\end{array}
$
\end{flushright}

\begin{lem}
\label{le5}
Let $T$, $v_j$ be as in Lemma~\ref{le4}.
Let $B$ be the square with vertices $( \pm 1 , \pm 1 )$.
Then for $n \ge 35$,
$$
B \subseteq \bigcup_{\ep_1 , \ep_2 , \dd , \ep_n \in \{ 0,1 \}}
\left[ \lt \sum_{j=1}^n \ep_j v_j \rt + \frac{1}{2} T^n B \right] ~.
$$
\end{lem}

\pf
$\frac{1}{2} T^n$ is the parallelogram with vertices
$$\pm \frac{1}{2} (1,-n) \pm \frac{1}{2} (0,1) ~.$$
The cross-section of this with $x$-coordinate $x_0$ is the vertical
interval $[-nx_0 - 1/2 , -nx_0 + 1/2 ]$ for $- 1/2 \le x_0 \le 1/2$.
Hence given $( \af , \be ) \in B$ pick $a \in \{ -1,0,1 \}$ such that
$- 1/2 \le \af + a \le 1/2$ and then pick $b \in \ZZ$ such that
$-n ( \af + a ) + 1/2 \le \be + b \le -n ( \af + a ) + 1/2$.
Since $| \beta | \le 1$ and $| \af + a | \le 1/2$,
we see $|b| \le \frac{n+1}{2} +1 \le n-16$ if $n \ge 35$.
Then $( \af , \be ) + (a,b) \in \frac{1}{2} T^n B$ and by Lemma~\ref{le4}
we can pick $\ep_1 , \dd , \ep_n$ such that
$\dis\sum_{j=1}^n \ep_j v_j = - (a,b)$, so Lemma~\ref{le5} follows. \hfill $\blacksquare$
\begin{prp}
\label{pr3}
$-1 \in {\rm int} \, \overline{W}$.
\end{prp}

\pf
Since $\overline{W}$ is closed under $z \mapsto \overline{z}$
and $z \mapsto 1/z$, it suffices to show that for $|z| <1$, ${\rm Im} \, z > 0$,
and $|z+1|$ sufficiently small, $z$ is in $\overline{W}$.
(Proposition~\ref{pr1} handles the case $z \in \RR$.)
Let $\dt = z+1$.
Let $B$ be the parallelogram with vertices $\pm 1 \pm \dt$.

We work in a nonstandard coordinate system for $\CC$, with basis vectors 1 and $\dt$, so $B$ is
represented by the square with vertices $(\pm 1 , \pm 1 )$.
We claim that multiplication by $z$ is represented by the matrix
$T = \mbox{\tiny $\lt \begin{array}{rr} -1 & 0 \\ 1 & -1 \end{array} \rt$}$ up to
$O( | \dt | )$.
We have
\begin{eqnarray*}
z \cdot 1 & = & -1 + \dt \\
z \cdot \dt & = & -\dt + \dt^2
\end{eqnarray*}
and
$$\dt^2 - 2 ( {\rm Re} \, \dt ) \dt + | \dt |^2 =0$$
so $\dt^2$ corresponds to $( | \dt |^2 , -2\, {\rm Re} \, \dt )$ in our basis,
and is $O( | \dt | )$.

From Lemma~\ref{le5}, it follows then that
$$B \subseteq \bigcup_{\ep_1 , \dd , \ep_{35} \in \{ 0,1\}}
\left[ \lt \sum_{j=1}^n \ep_j z^j \rt + \lt
\frac{1}{2} + O ( | \dt |) \rt z^n B \right]$$
so for sufficiently small $\dt$, we may apply Lemma~\ref{le1} to deduce
$z \in \overline{W}$. \hfill $\blacksquare$

We now combine all the results of this section.

\begin{theorem}
\label{t302}
There is an open neighborhood of $\{ z : |z| =1 , z \neq 1 \}$ contained in $\overline{W}$.
\end{theorem}

\pf
Apply Propositions~\ref{pr2} and \ref{pr3}. \hfill $\blacksquare$
\begin{cor}
\label{co1}
If $z \in (-1, -1 + \dt )$ for sufficiently small $\dt$ then $z$ is a multiple zero of some $0,1$ power series.
\end{cor}

\pf
By Theorem~\ref{t302},
if $\dt$
is small enough we can pick $0,1$ power series $f_n$ and zeros $z_n$ of $f_n$ such that
$z_n \not\in \RR$ and $z_n \to z$ as $n \to \In$.
By taking a subsequence we may assume that the coefficient of $z^k$ in $f_n$
is eventually constant for large $n$,
for each $k$.
By a Rouch\'{e}'s Theorem argument, the pairs of zeros
$\{ z_n , \overline{z}_n \}$ of $f_n$ must converge to (at least)
a double zero at $z$ of $\lim\limits_{n \to \In} f_n$. \hfill $\blacksquare$

\section{$\overline{W}$ is connected}
\hspace*{\parindent}
Since $W$ is countable, we cannot hope to prove $W$ is connected.
We prove instead
that $\overline{W}$ is connected.
First we need some topological lemmas.

Give $\{ 0,1 \}$ the discrete topology and $\{0,1\}^\omega$ the product topology,
as usual.
If $v = (v_1 , v_2 , \dd , v_n )$ is a finite vector of $0$'s and $1$'s, let
$S_v$ be the set of sequences in $\{0,1\}^\omega$
which start with $v$.
The following lemma is the key ingredient in the connectivity proof.
\begin{lem}
\label{le401}
Let $Y$ be a topological space.
Suppose $f : \{ 0,1 \}^\omega \to Y$ is a continuous map such that
\beql{eq401}
f(S_{v0} ) \cap f (S_{v1} ) \neq \emptyset
\eeq
for all $v \in \{ 0,1 \}^n$, and all $n \ge 0$.
(Here $v0$ denotes the vector $v$ with $0$ appended, etc.)
Then the image of $f$ is path connected.
\end{lem}

\pf
Let $w(0) = f( x'_0)$ and $w(1) = f(x_1)$ be elements of the image
we wish to connect by a path.
Find $x_{1/2}$, $x'_{1/2} \in \{0,1\}^w$ such that
$x'_0 , x_{1/2}$ have the same first coordinate,
and $x'_{1/2}$, $x_1$ have the same first coordinate and
$f(x_{1/2} ) = f(x'_{1/2} )$.
(If $x'_0 , x_1$ have the same first coordinate, take $x_{1/2} = x'_{1/2} = x'_0$;
otherwise apply the hypothesis \eqn{eq401} with $v$ as the empty vector.)
Let $w( 1/2)$ be this common value.

Next find $x_{1/4}$, $x'_{1/4} \in \{0,1 \}^w$,
using the same argument, such that $x'_0 , x_{1/4}$ agree in the first two
coordinates,
$x'_{1/4}$, $x_{1/2}$ agree in the first two coordinates, and
$f(x_{1/4} ) = f( x'_{1/4} )$.
Let $w(1/4)$ be this common value.
Do the analogous thing at 3/4.

By induction, we may continue to define $x_{d/2^n}$, $x'_{d/2^n}$,
$w(d/2^n )$ at all dyadic rationals
$d/2^n$ in $[0,1]$, such that
$x'_{d/2^n}$ and $x_{d+1/2^n}$ agree in the first $n$ coordinates
and
$$w(d/2^n) = f( x_{d/2^n}) = f(x'_{d/2^n} ) ~.$$
By induction, we see that all the $x'_q$ with
$q \in \left [ \frac{d}{2^n} , ~ \frac{d+1}{2^n} \right)$
agree in the first $n$ coordinates.
Hence for
$$r = \sum_{i=1}^\In \ep_i 2^{-i} \in [0,1] ~, ~~~
\ep_i \in \{0,1\}$$
not a dyadic rational, we may define
$$
x_r = x'_r = \lim_{n \to \In} x'_{\sum\limits_{i=1}^n \ep_i 2^{-i}}$$
and $w(r) = f(x_r)$.
Then $w$ maps
$\left[ \frac{d}{2^n} ,~ \frac{d+1}{2^n} \right]$ into $f(S_v)$ where
$v \in \{ 0,1 \}^n$ is the first $n$ coordinates of $x'_r$,
$r \in \left[ \frac{d}{2^n} ,~ \frac{d+1}{2^n} \right)$, and of
$x_{d+1/2^n}$.

We now show that $w$ is continuous at $r \in [0,1]$.
Let $U$ be an open set of $Y$ containing $w(r)$.
Then $f^{-1} (U)$ contains $S_v$ and $S_{v'}$ for some finite
substrings $v,v'$ of $x_r , x'_r$ respectively, by continuity of $f$.
By the last sentence of the previous paragraph it follows that
$$w^{-1} (U) \supseteq w^{-1} (f(S_v) \cup f(S_{v'} ))$$
will contain a neighborhood of $r$.

Thus $w: [0,1] \to \mbox{image}~ (f)$ is a continuous
path, and image $(f)$ is path connected. \hfill $\blacksquare$

Let $M$ be a topological space.
Give $M^n$ the product topology and let the symmetric group $S_n$ act on $M^n$
by permuting the coordinates.
The space $M^n / S_n$, which parameterizes $n$-element multisets, can be
given the quotient topology.

\begin{lem}
\label{le402}
If $A \subseteq M^n/ S_n$ is connected, and the multiset $\{ P, P, \dd , P \}$ is in $A$ for some $P \in M$, then the subset $B \subseteq M$ of all coordinates of points in $A$ is connected.
\end{lem}

\pf
Suppose not.
Then there are open sets $U$, $V \subseteq M$ such that
$U \cap B$ and $V \cap B$ are disjoint nonempty sets with union $B$.
Without loss of generality, $P \in U$.
Let
$$U' = U \times U \times \cdots \times U~,$$

\begin{eqnarray*}
V' & = & (V \times M \times M \times \cdots \times M ) \\
&~& \cup~ (M \times V \times M \times \cdots \times M ) \\
& ~ & ~~~~~~~~~~~~~~~~~\vdots \\
&~& \cup~ ( M \times M \times M \times \cdots \times V)~.
\end{eqnarray*}
Then $U'$, $V'$ are
open sets in $M^n$ which are
stable under $S_n$, so they project to
open sets
$U''$, $V''$ in $M^n /S_n$.
Also $A \subseteq U'' \cup V''$ since a point in $A$ must have all
coordinates in $U$,
or else at least one coordinate in $B \setminus U \subseteq V$.
Furthermore $P \in U'' \cap A$, and $V'' \cap A$ is nonempty also,
since at least one point of $A$ has a coordinate in $V$, since
$V \cap B \neq \emptyset$.
Finally $U'' \cap V'' \cap A = \emptyset$, since it is not possible
for a point of $A$ to have all coordinates in $U$,
yet have some coordinate in $V$.
This contradicts the connectedness of $A$. \hfill $\blacksquare$

\begin{theorem}
\label{t401}
$\overline{W}$ is connected.
\end{theorem}

\pf
First we show that for $\delta \in (0,1)$,
$$\overline{W}_\delta = ( \overline{W} \cap
\{ z : |z| \le 1 \} ) \cup \{
z: 1- \delta \le |z| \le 1 \} ~$$
is connected.
The idea is to apply Lemma~\ref{le401} to the function $f$ which assigns to
$( \ep_1 , \ep_2 , \dd )$ the set of zeros of
$$1 + \ep_1 z + \ep_2 z^2 + \cdots$$
inside $\{ z : |z| < 1 - \dt \}$.
To make a continuous map of this requires some manipulation.

By Jensen's theorem, as was shown in Section~2, there is an upper bound $n$ on the number of zeros that a power series
with $0,1$ coefficients can have
inside $\{ z: |z| < 1- \delta \}$.
Let $M$ be $\{ z: |z| \le 1 \}$ with the annulus
$\{ z: 1- \delta \le |z| \le 1 \}$ shrunk to a point $P$.
(Therefore $M$ is topologically a sphere.)
To each power series $1 + \sum\limits_{i=1}^\In \ep_i z^i$,
$\ep_i \in \{0,1\}$,
we assign the set of zeros inside $\{ z: |z| < 1- \delta \}$,
(counted with multiplicities) and throw in extra
copies of the point $P$ as necessary to bring the total number of points
to $n$.
Since the order of these $n$ elements of $M$ is unspecified, we obtain a point
of $M^n /S_n$.
Let $f((\ep_1 , \ep_2 , \dd ))$ be this point.

We claim that this map
$$f: \{ 0,1 \}^\omega \to M^n / S_n$$
is continuous.
This follows easily from Rouch\'{e}'s theorem;
if two power series agree in the first $m$ coordinates for $m$
sufficiently large then their zeros inside
$\{ z: |z| < 1- \delta \}$ will be within $\ep$.
Some may escape or enter the disk, but this is not a problem, since in the topology on $M$,
$P$ is close to all points
$z$ with $|z|$ sufficiently near $1- \delta$.

We next check condition \eqn{eq401} of Lemma~\ref{le401}.
This is easily done using the following trick:
given $v = (v_1 , v_2 , \dd , v_n ) \in \{ 0,1 \}^n$, let
$w = (v_1 , v_2 , \dd , v_n , 1, v_1 , v_2 , \dd , v_n )$.
Then $v \in S_{v0}$,
$w \in S_{v1}$, and $f(v) = f(w)$ (we extend $v,w$ to infinite
vectors by appending $0$'s), since
$$1 + v_1 z + v_2 z^2 + \cdots + v_n z^n$$
and
$$
\begin{array}{l}
1+ v_1 z + v_2 z^2 + \cdots + v_n z^n + z^{n+1} + v_1 z^{n+2} + \cdots + v_n z^{2n+1} \\
~~ \\ [-.09in]
~~~~~~~~=~ (1+ v_1 z + v_2 z^2 + \cdots + v_n z^n ) (1+ z^{n+1} )
\end{array}
$$
have the same zeros inside $\{ z : |z| < 1- \delta \}$.
Therefore we may apply Lemma~\ref{le401} and deduce that the image
of $f$ is path connected.

Since $f((0,0, \dd )) = (P,P,P, \dd , P)$,
we may apply Lemma~\ref{le402} with $A = \mbox{image} (f)$ to deduce that
$\overline{W}_\delta$ with the annulus
$\{ z: 1- \delta \le |z| \le 1 \}$ shrunk to a point $P$ is a connected
subset of $M$.
This is equivalent to the connectivity of $\overline{W}_\delta$.

Since $\overline{W} \cap \{ z : |z| \le 1 \}$ is the decreasing
intersection of the compact connected sets $\overline{W}_{1/m}$,
it too is connected.
So is its image under $z \mapsto 1/z$.
Finally, $\overline{W}$ is the union of these two sets,
which meet on the unit circle, so $\overline{W}$ is connected as well. \hfill $\blacksquare$
\section{$\overline{W}$ is path connected}
\hspace*{\parindent}
Here we refine the argument of the previous section to prove
$\overline{W}$ is path connected.
There are two main difficulties that arise.
One is that the path connected analogue of Lemma~\ref{le402},
although still true (at least when $M$ is Hausdorff), is much
harder to prove.
The second is that a decreasing intersection of compact path
connected sets need not be path connected, so we can no longer
restrict our attention to the zeros within
$\{ z: |z| < 1 - \dt \}$.

The lifting lemma below will be used as a substitute for
Lemma~\ref{le402}.
Its proof is based on proofs obtained independently by David
desJardins and Emanuel Knill.
\begin{lem}
\label{le501}
{\rm (Lifting lemma):}
Let $M$ be a Hausdorff space and let $\pi: M^n \to M^n /S_n$ be
the projection map.
Let $f: [0,1] \to M^n/S_n$ be a continuous map.
Then there is a continuous map $g: [0,1] \to M^n$ such that $f =
\pi \circ g$.
\end{lem}
\begin{sle}
\label{sl501}
Let $\Delta = \{ t \in [0,1]: f(t)$ consists of $n$ copies of a
single point$\}$.
Let $g: [0,1] \to M^n$ be an arbitrary function that is a lift
of $f$.
Then $g$ is automatically continuous at all $t_0 \in \Delta$.
\end{sle}

\pf
Suppose $t_0 \in \Delta$ and $f(t_0) = \{ x,x, \dd , x \}$.
If $U$ is an open neighborhood of $x$,
$$g^{-1} (U^n) = f^{-1} ( \pi (U^n))$$
which is open.
Since such subsets $U^n$ form a neighborhood base at
$(x,x, \dd , x) \in M^n$, this proves that $g$ is continuous at
$t_0$.
\begin{sle}
\label{sl502}
Let $I_1$, $I_2$ be closed subintervals of $[0,1]$ such that
$I_1 \cap I_2$ is a single point $\{t \}$.
If $g_j$ is a continuous lift of $f$ on $I_j$ $(j=1,2)$ then
there is a continuous lift $g$ of $f$ on $I_1 \cup I_2$ such
that $g|_{I_1} =g_1$.
\end{sle}

\pf
Since $g_1 (t)$ and $g_2 (t)$ differ only by a permutation, we
can compose $g_2$ with a permutation $\sigma: M^n \to M^n$ and
then paste the result to $g_1$.
\begin{sle}
\label{sl503}
The conclusions of Sublemma~\ref{sl502} hold even if $I_1$ and $I_2$ intersect in more than a
point.
\end{sle}

\pf
This follows form Sublemma~\ref{sl502} since $I_1 \cup I_2$ can
be expressed as the union of $I_1$ with at most two closed
subintervals of $I_2$ each meeting $I_1$ in a point.
\begin{sle}
\label{sl504}
If $I$ is a closed subinterval of $[0,1]$ and every $t \in I$
has a neighborhood on which $f$ has a lift, then $f$ has a lift
on $I$.
\end{sle}

\pf
By compactness, we can cover $I$ by closed intervals $I_1 , I_2
, \dd , I_k$ on which $f$ has a lift, and we may assume
$I_j \cap I_{j+1} \neq \emptyset$ for $1 \le j \le k$.
By induction on $j$, Sublemma~\ref{sl503} lets us extend the
lift on $I_1$ to a lift on $I_1 \cup I_2 \cup \cdots \cup I_j$.
\begin{sle}
\label{sl505}
The same holds if $I$ is any subinterval of $[0,1]$.
\end{sle}

\pf
Let $C_1 \subseteq C_2 \subseteq \cdots$ be closed intervals
such that $\bigcup\limits_{i=1}^\In C_i = I$.
By Sublemma~\ref{sl504}, there is a lift on each $C_i$.
By repeated use of Sublemma~\ref{sl503}, extend the lift on
$C_1$ to a lift on $C_2$, extend this to $C_3$, etc.
This process gives a lift on $I$.

\vsp
\noindent{\bf Proof of Lemma~\ref{le501}.}
We use induction on $n$.
The case $n=1$ is trivial, so assume $n > 1$.
By Sublemma~\ref{sl501}, it suffices to find a lift
on each connected component $I$ of $[0,1] \setminus \Delta$.
By Sublemma~\ref{sl505} it suffices to show that any
$t_0 \in I$ has a neighborhood on which there is a lift.

Suppose $z_1 , z_2 , \dd , z_k$  ($k \ge 2$) are the distinct elements of the
multiset $f(t_0)$,
occurring with multiplicities $n_1 , n_2 , \dd , n_k$
respectively.
Since $M$ is Hausdorff, there exist pairwise disjoint
neighborhoods $U_i$ of $z_i$.
Let $N$ be a closed interval neighborhood of $t_0$ such that
$t \in N$ implies $f(t) \in \pi ( U_1^{n_1} \times \cdots \times
U_k^{n_k})$.
Then on $N$, we can lift $f$ to a path $\tilde{f}$ in $M^{n_1}
/S_{n_1} \times \cdots \times M^{n_k} / S_{n_k}$ since the
projection
$$M^{n_1} / S_{n_1} \times \cdots \times M^{n_k} / S_{n_k} \to
M^n / S_n$$
restricts to a homeomorphism
on the projections of $U_1^{n_1}
\times \cdots \times U_k^{n_k}$.
By the inductive hypothesis applied to each of the $k$
coordinates of $\tilde{f}$, we can lift $\tilde{f}$ to a path in
$M^{n_1} \times \cdots \times M^{n_k} = M^n$ as desired. \hfill $\blacksquare$
\begin{theorem}
\label{th501}
$\overline{W}$ is path connected.
\end{theorem}

\pf
Let $M$ be $\{ z : |z| \le 1 \}$ with the unit circle shrunk to
a point $P$.
Again $M$ is topologically a sphere,
so we may give it a bounded metric $d$.
Let $M^{\In}$ be the set of sequences $x = \{ x_i \}_{i=1}^\In$
which converge to $P$ and define a metric $d_\In$ on $M_\In$ by
$$d_\In (x,y) = \sup_i d(x_i , y_i ) ~.$$
Let the group $S_\In$ of permutations of $\{1,2,\dd , \}$ act on
$M^\In$ by permuting the coordinates.
Define a metric $D$ on the quotient space $M^\In / S_\In$ by
letting
$$D( \overline{x} , \overline{y} ) = \inf_{\sigma \in S_\In}
d_\In (x, \sigma y) ~.$$
Here $( \overline{x} , \overline{y} )$ denote the projections of
$x,y \in M^\In$ to $M^\In / S_\In$.
(That $D( \overline{x} , \overline{y} ) =0$ if and only
if $\overline{x} = \overline{y}$ requires the convergence of
$x,y$.)
The set of zeros of a power series
$$1+ \ep_1 z + \ep_2 z^2 + \cdots$$
inside $\{ z: |z| < 1 \}$ forms a sequence in $M$ converging to
$P$
(by Proposition~\ref{pr201}) or else is finite, in which case we
append an infinite sequence of $P$'s.
This defines a map
$$f : \{ 0,1 \}^\omega \to M^\In /S_\In ~.$$
By the same Rouch\'{e}'s theorem argument used in the proof of
Theorem~\ref{t401}, this map is continuous.
The conditions of Lemma~\ref{le401} hold for the same reason as
before, so the image of $f$ is path connected.

Suppose $z_0 \in \overline{W} \cap \{ z: |z| < 1 \}$.
Let $\omega : [0,1] \to M^\In / S_\In$ be a path from the image
under $f$ of a $0,1$ power series vanishing at $z_0$ to
$f((0,0, \dd )) = \{ P,P,P, \dd \}$.

Fix $m \ge 1$, and let $M_m$ be $\{ z: |z| \le 1 \}$ with the
annulus $\{ z : 1 - 1/m \le |z| \le 1 \}$
shrunk to a point $Q$.
Define $|~|$ on $M_m$ by letting $|Q| =1 - 1/m$.
By Proposition~\ref{pr201} there is an upper bound $n$ on the
number of zeros of a $0,1$ power series inside $\{z: |z| < 1 -
1/m \}$.
The path $\omega$ induces a path
$$\om_m : [0,1] \to (M_m)^n /S_n ~.$$
(Apply the projection $M \to M_m$ to each element of $\om (t)$,
and throw away infinitely many $Q$'s to get $\om_m (t)$.)

Pick $m_0 \ge 1$ such that $|z_0| < 1 - \frac{2}{m_0}$.
We define inductively a sequence of paths
$$\tilde{\om}_m : [0,t_m ] \to \overline{W} ~,~~~
m=m_0 , m_0 + 1 , \dd ~,$$
each extending the one before.
First apply Lemma~\ref{le501} to lift $\om_{m_0}$ to a path
$[0,1] \to M_{m_0}^n$.
Since some coordinate of $\omega_{m_0} (0)$ is $z_0$ and since
all
coordinates of $\om_{m_0} (1)$ are $Q$, we get a path
$\tilde{\om}_{m}$ from $z_0$ to $Q$ in $M_{m_0}$.
Let $t_{m_0}$ be the smallest $t \in [0,1]$ such that
$| \tilde{\om}_{m_0} (t) | \ge 1- 2/ m_0$.
Then by restriction to $[0, t_{m_0} ]$ we get a path
$\tilde{\om}_{m_0}$ in $\CC$ since
$\{ z \in M_{m_0} : |z| \le 1- 2/ m_0 \}$ can be identified with
$\{ z \in \CC : |z| \le 1 - 2 / m_0 \}$.
Finally, since $\tilde{\om}_{m_0} (t)$ is always a coordinate of
$\om (t)$, $\tilde{\om}_{m_0} (t) \in \overline{W}$ for all $t
\in [0, t_{m_0} ]$.

By the same process, we inductively find for each $m > m_0$ a path
$\tilde{\om}_m : [t_{m-1} , 1 ] \to M_m$ such that
$\tilde{\om}_m (t_{m-1} ) = \tilde{\om}_{m-1} (t_{m-1} )$, let
$t_m$ be the smallest $t \ge t_{m-1}$ such that
$$| \tilde{\om}_m (t) | \ge 1 - \frac{2}{m}~,$$
and obtain a path
$$\tilde{\om}_m : [ t_{m-1} , t_m ] \to \overline{W}$$
which we append to $\tilde{\om}_{m-1}$ to obtain
$$\tilde{\om}_m : [0, t_m ] \to \overline{W}$$
such that $\tilde{\om}_m (t)$ is always a coordinate of $\omega (t)$.

Let $t_\In = \dis\sup_{m} t_m$.
Piecing together the $\tilde{\om}_m$'s gives a continuous map
$$\tilde{\om} : [0, t_\In ) \to \overline{W}$$
such that $\tilde{\om} (t)$ is a coordinate of $\om (t)$ for all
$t \in [0, t_\In )$.
The set of limit points of $| \tilde{\om} (t) |$ as $t \to
t_\In$ is a closed interval $I$.
Let $\om (t_\In ) = \{ z_1 , z_2 , z_3 , \dd \}$.
If $r \in [0,1)$ is distinct from $|z_1 |, |z_2|, |z_3 | , \dd$ then $\dis\sup_i \left| r- |z_i | \right| > 0$ (since
$\{ z_i \} \to P$) and by continuity of $\om$, $r$ also
differs by some $\ep$ from all coordinates of $\om (t)$ for $t$
in a neighborhood of $t_\In$, so $r$ cannot be a limit point of
$| \tilde{\om} (t) |$ as $t \to t_\In$.
Thus $I \subseteq \{ 1, |z_1 | , |z_2 | , \dd \}$ but $|z_i |
\to 1$ so $I$ must be a single point.
Since $| \tilde{\om} (t_m) | =1 - 2/m$ for $m \ge m_0$,
$\lim\limits_{t \to t_\In} | \tilde{\omega} (t) | =1$.

\vsp
\noindent{\bf Case~1:}
1 is the only limit point of $\tilde{\om} (t)$ as $t \to t_\In$.
Then $\tilde{\om}$ extends to a path $[0, t_\In] \to
\overline{W}$ from $z_0$ to 1.

\vsp
\noindent{\bf Case~2:}
There is a limit point $\theta \neq 1$, $| \theta | =1$, of
$\tilde{\om} (t)$ as $t \to t_\In$.
By Theorem~\ref{t302}, there is an open disc centered at
$\theta$ contained in $\overline{W}$.
For some $t < t_\In$, $\tilde{\om} (t)$ is in this disc, so we
can replace the tail end of $\tilde{\om}$ on $[t, t_\In )$ by a
straight line from $\tilde{\om} (t)$ to $\theta$ in
$\overline{W}$.

In either case we can connect $z_0$ to a point on the unit
circle via a path in $\overline{W}$.
The same is true if $z_0 \in \overline{W}$,
$|z_0| > 1$, since $\overline{W}$ is closed under $z \mapsto 1/z$.
Since $\overline{W}$ contains the unit circle, this proves that
$\overline{W}$ is path connected. \hfill $\blacksquare$

\section{Graphs, computations, and the shape of $\overline{W}$}
\hspace*{\parindent}
The computations of zeros graphed in our figures were performed
in double precision (approx. 18 decimal places) on a Silicon
Graphics workstation.
Some of the zeros were checked for accuracy by recomputing them
in double precision (approx. 28 decimal places) on a Cray X-MP.
The zero-finding program used the Jenkins-Traub algorithm and
was taken from a standard subroutine library.
Checks showed that the values that were obtained were accurate
on average to at least 10 decimal places, which was sufficient
for our graphs.
The program that was used appeared to produce accurate values on
the Cray for the zeros
for polynomials of degrees up to about 150.
(Computation of zeros of polynomials of much higher degree would
have required better algorithms, cf.~\cite{CC}.)

Zeros of a large set of random polynomials $f(z) \in P$ of degree 100 were computed on the Cray,
and they exhibit most of the features visible in Figures~1--3.
However, they are not as interesting as the lower degree zeros that are exhibited in Figures~1--3.
The ``spikes'' or ``tendrils'' that generate the fractal appearance in the graphs we include come from a small fraction of the
polynomials.
Sampling even $10^4$ of the $2^{99}$ polynomials
$f(z) \in P$ of degree 100 does not yield a good representation of the extremal features that we expect to see
for high as well as low degrees.

Graphs were prepared using the S system \cite{BCW}.

The graphs in Figures~4--6 were prepared differently.
A program was written that checked whether a given $w$ with
$|w| < 1$ is in $\overline{W}$.
Note that
\beql{eq601}
\left| \sum_{k=0}^\In a_k w^k \right| \le B =
\max (1, | 1+w|) / (1- |w|^2 ) ~,
\eeq
where the $a_k$ are any $0,1$ coefficients,
since we can write
$$\sum_{k=0}^\In a_k w^k = (a_0 + a_1 w) + (a_2 + a_3 w) w^2 +
\cdots ~.$$
The procedure was to test all sets of $0,1$ coefficients
$a_1 , \dd , a_{120}$ to see whether they could be the initial segment of coefficients of a power series
\beql{eq602}
f(z) = 1 + \sum_{k=1}^\In a_k z^k
\eeq
for which $f(w) =0$.
Let us regard the strings of coefficients $a_1 , \dd , a_{120}$ as the
leaves of a balanced binary tree, with the nodes below the root corresponding to
$a_1$, those below to $a_1 , a_2$, etc.
The procedure was to explore this tree,
checking whether
\beql{eq603}
\left| 1 + \sum_{j=1}^d a_j w^j \right| > |z|^{d+1} B
\eeq
at any stage.
If \eqn{eq603} is satisfied, then $w$ is not a zero of any power series of
the form \eqn{eq602} with initial coefficients
$1, a_1 , \dd , a_d$, and the subtree of that node does not have to be explored.
If all the leaves are discarded by this procedure, we have a rigorous
proof that $w \not\in \overline{W}$, and so in fact an open neighborhood of $w$ is outside $\overline{W}$.
On the other hand, if a leaf was found with
\beql{eq604}
\left| 1 + \sum_{j=1}^{120} a_j w^j \right| < |z|^{121} B/10 ~,
\eeq
then the program assumed that $w \in \overline{W}$.
(By establishing lower bounds for the derivative of the polynomial
$1 + \sum\limits_1^{120} a_j z^j$ at $w$ and using crude upper bounds for
the second derivative, one could in principle prove that there is some point $w'$ close to $w$ such that $w ' \in \overline{W}$, although the 10
in condition \eqn{eq604} might have to be decreased.
Another way to prove this would be to use Lemma~\ref{le1}.
This step was not carried out.)
Figures~4---6 were produced by testing each $w$ in a $1936
\times 1936$
or a $1944 \times 1944$
grid (corresponding to the resolution of our laser
printer).
There were few points $w$ for which neither condition
\eqn{eq603} nor condition \eqn{eq604} held.
The exceptions occur primarily in Fig.~4, but they do not affect
how the picture looks.
Had we used a tree of depth 80, the exceptions would have been
much more frequent.

The computations of Figures~4--6 are not completely rigorous in
that the determination of $w \not\in \overline{W}$ is rigorous,
while that of $w \in \overline{W}$ is not.
Moreover, an implicit premise in the preparation of Figures~4--6
was that if a point $w \in \overline{W}$, then the whole
neighborhood of $w$ represented by the corresponding pixel is in
$\overline{W}$.
On the other hand, the computations of Figures~1--3 are
rigorous.

It is possible to use computations to obtain rigorous
estimates for $\overline{W}$ that are sharper than those of
Theorem~\ref{t201}.
As an example, we sketch how a moderate amount of straightforward computing
establishes that there are no $w \in \overline{W} \setminus \RR$
with $|w| < 0.7$.
We modify the method of proof of Theorem~\ref{t201}.
Write
\beql{AMO65}
f(z) = 1 + \sum_{j=1}^{10} a_j z^j + \frac{1}{2}
(z^{11} + z^{12} + \cdots ) + g(z) ~,
\eeq
where
\beql{AMO66}
g(z) = \frac{1}{2} \sum_{k=11}^\In \ep_k z^k , ~~~~
\ep_k = \pm 1 ~.
\eeq
Then we can write
\begin{eqnarray*}
f(z) & = & F(z) + g(z) ~, \\
F(z) & = & \frac{G(z)}{2(1-z)} ~, \\
G(z) & = & 2(1-z) \lt 1 + \sum_{j=1}^{10} a_j z^j \rt + z^{11}
~.
\end{eqnarray*}
If we establish that $|F(z)| > |g(z)|$ on some simple closed
contour about the origin, then by Rouch\'{e}'s theorem $f(z)$
and $F(z)$ will have the same number of zeros inside that
contour.
To prove that $|F(z)| > |g(z)|$ on a contour $C$,
it suffices to show that $|F(z) | > g(z) + \delta$ on a discrete
set of points $z$ on $C$, where
$\delta > 0$ is such that bounds on the derivatives of $F(z)$
and $g(z)$ guarantee that $|F(z)| - |g(z)|$ will not decrease by
more than $\delta$ between the sampling points.
This was applied to each of the $2^{10}$ choices of
$a_1, \dd , a_{10}$.
Of the 1024 functions $F(z)$, 997 satisfied $|F(z) | > |g(z)|$
on
$$C_3 = \{ z: |z| = 0.7 \}~.$$
The remaining 27 functions $F(z)$ were shown to satisfy
$|F(z)| > |g(z)|$ on the contour
\begin{eqnarray*}
\lefteqn{C_4 = \{ z: |z| = 0.7, ~ |y| \ge 0.04 \}} \\
&&~~~~\cup \, \{ z: x = - 0.74, ~ |y| \le 0.04 \} \\
&&~~~~\cup \, \{ z: |y| = 0.04, ~ -0.74 \le x \le -0.6 , ~
|z| \ge 0.7 \} ~.
\end{eqnarray*}
Finally, zeros of each of the 1024 polynomials $G(z)$ were
computed, and it was found that 85 of these polynomials had a
single zero in $|z| \le 0.74$, and the remaining 939 had none.
Thus in all cases we can conclude that $f(z)$ has at most one
zero in $|z| \le 0.7$.
Such a zero has to be real.

The estimates used above were crude, and with more care one
can either decrease the amount of computing (and even eliminate
it altogether) or obtain better bounds for $\overline{W}$.

The basic principle that makes it possible to obtain good
estimates of $\overline{W}$ is that for extremal points $w \in
\overline{W}$, the power series $f(z)$ with
$0,1$ coefficients such that $f(w) =0$ are restricted.
For example the region depicted in Figures~3 and 4 is
$$V = \{ z = x+ iy : -0.501 \le x \le - 0.497 ,~
0.537 \le y \le 0.541 \} ~.$$
Numerical computation (evaluating polynomials of degrees $\le 9$
with $0,1$ coefficients at a $41 \times 41$ uniform grid, and
bounding derivatives) shows that if $w \in V \cap \overline{W}$,
then $w$ can only be a zero of a power series of the form
$$f(z) = 1+ z + z^2 + z^4 + z^7 + z^9 + \sum_{k=10}^\In a_k
z^{10} ~.$$
This restricted the set of $f(z)$ that had to be considered, and made
possible the computation of Figure~3, as it would not have been
feasible to examine all polynomials of degrees $\le 32$.
Furthermore, this restriction on the coefficients of $f(z)$
makes it possible to estimate the shape of $V \cap
\overline{W}$.

It should be possible to prove rigorously, with the help of numerical computations, such as those mentioned above, that the hole in $\overline{W}$ mentioned in the Introduction and pictured in Figure~6 is isolated in the
sense that there is a continuous closed curve in $\overline{W}
\cap U$, for $U$ a small rectangle, that encloses the hole.
We have not done this.

To explain the fractal appearance of $\overline{W}$,
suppose that $w \in W$, $|w| < 1$, and that
$f(w) =0$ where
$$f(z) = 1 + \sum_{j=1}^d a_j z^j ~, ~~~~a_j = 0,1 ~.$$
Suppose that
$$g(z) = f(z) + z^{d+1} \sum_{k=0}^\In b_k z^k ~, ~~~~ b_k = 0,1
~.$$
If $g(z) =0$ and $|z-w|$ is small, while $d$ is large, we have
\begin{eqnarray*}
0 ~=~ g(z) & \cong & g(w) + (z-w) g'(w) \\
& \cong & w^{d+1} \sum_{k=0}^\In b_k w^k +
(z-w) f'(w) ~.
\end{eqnarray*}
If $f' (w) \neq 0$ (which  as far as we know may hold for all
$w$ with $|w| < 1$),
then $g' (w) \neq 0$ for $d$ large enough, and we can expect
that
$$z \cong w - \frac{w^{d+1} \sum_{k=0}^\In b_k w^k}{f'(w)} ~.$$
Thus if
$$Q(w) = \left\{ \sum_{k=0}^\In b_k w^k :
b_k =0,1 \right\} ~,$$
then we expect to find zeros in a neighborhood of each point of
$$w - w^{d+1} (f'(w))^{-1} Q(w) ~.$$
The set $Q(w)$ is connected \cite{B}, and for $w \not\in \RR$,
it seems that it contains a small disk around the origin.
The set $Q(w)$ is a continuous function of $w$, which accounts
for the similarity of the protrusions
from $\overline{W}$ visible in Figures~5 and 6.
(The protrusions in Fig.~4 are different,
since there the sets $Q(w)$ are of different shape from those in
Figures~5 and 6.)

\vsp
\noindent{\bf Acknowledgements}.
The authors thank E.~R. Rodemich for originally raising the
question of the distribution of zeros of $0,1$ polynomials, 
M. Sambandham for providing a copy of \cite{SB},
A.~R. Wilks for help with graphics, and D. Zagier for informing
them of the work of T. Bousch \cite{Bo1,Bo2}.
Special thanks are due to David desJardins and Emanuel Knill
for permission to use their proofs of
Lemma~\ref{le501}.

\clearpage
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\clearpage
\section*{Figure Captions}
\begin{list}
{}{\setlength{\leftmargin}{0.6in}\setlength{\labelwidth}{0.5in}
\setlength{\labelsep}{.1in}\hfill}
\item[Fig.~1.]
Scatterplot of zeros $z = x+ iy$ of polynomials of degrees $\le 16$ with constant term 1 and coefficients 0 and 1.
The ``segments'' along the negative real axis are created by
negative real zeros.
Negative real zeros $< - 1.5$ are not shown.
\item[Fig.~2.]
Scatterplot of zeros of polynomials of degree 18 with constant
term 1 and coefficients 0 and 1
that are near to $z=1$.
\item[Fig.~3.]
Scatterplot of zeros of polynomials of degrees $\le 32$ with
coefficients 0 and 1.
\item[Fig.~4.]
Section of $\overline{W}$.
The same region, with points from $W$ displayed, is shown in
Fig.~3.
Black denotes points $z \in \overline{W}$.
\item[Fig.~5.]
Section of $\overline{W}$, the set of zeros of power series with
$0,1$ coefficients with black denoting $z \in \overline{W}$.
\item[Fig.~6.]
Section of $\overline{W}$.
This is an enlargement by a factor of 80 of a section of Fig.~5, showing some of the holes contained in $\overline{W}$.
\end{list}
\end{document}