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Schr\"odinger operator, finite gap potential, WKB asymptotics, 
singular spectrum
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\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\newtheorem{Theorem}{Theorem}[section]
\newtheorem{Proposition}[Theorem]{Proposition}
\newtheorem{Lemma}[Theorem]{Lemma}
\newtheorem{Definition}[Theorem]{Definition}
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\begin{document}
\title{Finite gap potentials and WKB asymptotics for
one-dimensional Schr\"odinger operators}
\author{Thomas Kriecherbauer$^1$
and Christian Remling$^2$}
\date{February 26, 2001}
\maketitle
\begin{center}
(to appear in {\it Commun.\ Math.\ Phys.})
\end{center}
\vspace{0.5cm}
\noindent
1.\ Universit\"at M\"unchen,
Mathematisches Institut, Theresienstr.\ 39, 80333 M\"unchen, GERMANY\\
E-mail:  tkriech@rz.mathematik.uni-muenchen.de\\[0.1cm]
2.\ Universit\"at Osnabr\"uck,
Fachbereich Mathematik/Informatik,
49069 Osnabr\"uck, GERMANY\\
E-mail:
cremling@mathematik.uni-osnabrueck.de\\[0.3cm]
2000 AMS Subject Classification: primary 34L40, 81Q10,
secondary 30F99 \\[0.3cm]
Key words: Schr\"odinger operator, finite gap potential,
singular spectrum, Jacobi inversion problem\\[0.3cm]
\begin{abstract}
Consider the Schr\"odinger operator $H=-d^2/dx^2+V(x)$ with
power-decaying potential $V(x)=O(x^{-\alpha})$. We prove that
a previously obtained dimensional bound on exceptional sets
of the WKB method
is sharp in its whole range of validity.
The construction relies on pointwise bounds on finite
gap potentials. These bounds are obtained by an
analysis of the Jacobi inversion problem on hyperelliptic
Riemann surfaces.
\end{abstract}
\section{Introduction}
We are interested in one-dimensional Schr\"odinger equations,
\begin{equation}
\label{se}
-y''(x)+V(x)y(x)=Ey(x),
\end{equation}
and the spectra of the corresponding self-adjoint operators
$H_{\beta}=-d^2/dx^2 + V(x)$ on $L_2(0,\infty)$, say. The index
$\beta\in [0,\pi)$ refers to the boundary condition
$y(0)\cos\beta + y'(0)\sin\beta = 0$. The spectral properties
of the operators $H_{\beta}$ give information on the large time
behavior of the quantum mechanical system described by
\eqref{se}.

In this paper, we will present an alternate approach to
an earlier result of one of us \cite{Remex}.
With this new approach, we can
remove a technical condition and thus prove that a
previously obtained bound on the embedded singular spectrum
of $H_{\beta}$ is
sharp in its whole range of validity. We will describe this
result shortly; let us first point out that
the new idea of this paper is to use finite gap potentials
in the construction of \cite{Remex}.
The main difficulty is to
obtain good pointwise bounds on these potentials.
A substantial part of this paper is devoted to
this problem. More specifically, we will have to study in some detail
the Jacobi inversion problem on hyperelliptic Riemann
surfaces. The result of this analysis
is formulated as Theorem \ref{T3.2}.
Actually, our proof gives more than stated: We obtain a whole
sequence of good pointwise approximations (where, very
roughly, ``good'' means better than expected, due to cancellations)
to finite gap potentials. While our motivation for proving
Theorem \ref{T3.2} is to provide tools
for the proof of
Theorem \ref{T1.1} below, this discussion is perhaps of independent
interest.

Let us now return to \eqref{se};
suppose that the potential $V$ is bounded by a decaying
power, that is,
\begin{equation}
\label{hyp}
|V(x)| \le \frac{C}{(1+x)^{\alpha}} \quad\quad (\alpha > 0).
\end{equation}
Then, if $\alpha > 1/2$, the operators $H_{\beta}$ have absolutely
continuous spectrum essentially supported by $(0,\infty)$, as was first
proved in
\cite{CK,Remac}. Embedded
singular spectrum can occur if $\alpha \le 1$ (see
\cite{Nab,Simpp,vNW}), but there are restrictions on the dimension
of the singular part of the spectral measure. This
is intimately related to the problem of solving \eqref{se}
asymptotically (for large $x$). We say that a solution $y(x,E)$ of
the Schr\"odinger equation satisfies the WKB asymptotic formulae if
\begin{equation}
\label{WKB}
\left(\begin{array}{c}
y(x,E) \\ y'(x,E)
\end{array} \right) =
\left( \begin{array}{c} 1 \\  i\sqrt{E} \end{array} \right)
\exp\left( i\int_0^x \sqrt{E-V(t)}\, dt
\right) + o(1) \quad\quad (x\to\infty).
\end{equation}
It is well known that there exist solutions of \eqref{se}
satisfying \eqref{WKB} for all
$E>0$ if the potential $V$ decays and is slowly varying in
a suitable sense (see, for
instance, \cite[Chapter 2]{East1}). Obviously, this latter assumption
need not hold if $V$ only satisfies \eqref{hyp}.
Nevertheless, recent work
\cite{CK,CK1,CK2,Remac,Remdim}
has shown that \eqref{WKB} continues to hold off a small
exceptional set of energies $E$ as long as $\alpha>1/2$.
Call this exceptional set $S$; in other words,
\begin{equation}
\label{defS}
S=\{ E>0: \text{ No solution of \eqref{se} satisfies \eqref{WKB}} \} .
\end{equation}
General criteria \cite{GP,Sbdd,Stolz} show that if there is some
embedded singular spectrum on $(0,\infty)$, then the corresponding parts of
the spectral measures are supported by $S$. In other words, if
$\rho^{(\beta)}$ denotes the spectral measure of $H_{\beta}$, then
$\rho_{sing}^{(\beta)}((0,\infty)\setminus S)=0$ for all $\beta$.
Therefore, it is interesting to study $S$ in detail.
We know from \cite{CK,Remac} that $S$ is of Lebesgue measure
zero if $\alpha>1/2$; this was subsequently strengthened in
\cite{Remdim} where it was proved that the Hausdorff dimension
of $S$ satisfies $\dim S\le 2(1-\alpha)$. Formally, this result is valid
for all $\alpha\in\mathbb R$ (if one defines $\dim\emptyset=-\infty$),
but it gives nontrivial information only if $1/2<\alpha\le 1$.
We will show that
this bound is sharp and is even attained for suitable potentials:
\begin{Theorem}
\label{T1.1}
For every $\alpha\in (1/2,1]$,
there exist potentials $V(x)$ satisfying \eqref{hyp},
so that $\dim S= 2(1-\alpha)$.
\end{Theorem}
If $\alpha\notin (1/2, 1]$, the whole picture is different.
More precisely, if $\alpha\le 1/2$, then $S$ can have full Lebesgue
measure in $(0,\infty)$ \cite{KLS,KotU,Sim82} and the
spectrum can be purely singular. On the other hand, it is easy
to prove that $S=\emptyset$ if $\alpha>1$ (see, e.g., \cite{East1}).

In \cite{Remex}, Theorem \ref{T1.1}
was proved for $\alpha >2/3$. Things get
more difficult as $\alpha$ approaches $1/2$. In particular,
we really need the full force of
Theorem \ref{T3.2} in that the exponent $N$ there gets larger
and larger
as $\alpha$ decreases to $1/2$.

Actually, here, too, we show more
than stated: For any given function $\epsilon(x)$ with
$\epsilon(x)\to 0$ as $x\to\infty$
(no matter how slowly), we can construct
a potential $V(x)=O(x^{-\alpha-\epsilon(x)})$, so that
$\dim S=2(1-\alpha)$.

There are extensions of the results quoted
above to far more general settings. Deift and Killip \cite{DK}
have proved that there is absolutely continuous spectrum essentially
supported by $(0,\infty)$ already if $V\in L_1+L_2$; very recently,
Killip has obtained even stronger results in this direction
\cite{Kil}. WKB asymptotics off exceptional sets have been established
by Christ and Kiselev \cite{CK1,CK2} under
very general conditions, including $V\in L_1+L_p$ for some $p<2$
(but not in the borderline case $p=2$, which remains open).

A major open question in this context is Simon's problem no.\ 7
\cite{Sim21}: Are there potentials satisfying \eqref{hyp} with
$\alpha>1/2$, so that for some boundary condition $\beta$, the
operator $H_{\beta}$ has some singular {\it continuous} spectrum?

We organize this paper as follows. In Sect.\ 2, we discuss
the construction of the so-called finite gap potentials, that is,
of quasi-periodic potentials with finitely many prescribed
gaps in the spectrum. Since this material is rather classical,
we concentrate on those aspects of the theory that are needed later.
The following section introduces the problem of obtaining pointwise
bounds on finite gap potentials. We
state our main result on finite gap potentials (Theorem
\ref{T3.2}) and discuss some general features of this result.
The proof is given in Sect.\ 4, 5, 6; this analysis
is perhaps the central part of this paper. It depends on a study
of the Jacobi inversion problem in cases where a large number of
small gaps is present.
A major role will be played by a graphical representation of the
terms of a perturbation series, which we introduce in Sect.\ 5.
With Theorem \ref{T3.2}
as new input, we can then obtain Theorem \ref{T1.1}, relying
mainly on the ideas
already contained in \cite{Remex}. This is done in Sect.\ 7. In fact,
with our new approach, the treatment becomes more transparent.

{\bf Acknowledgment:}
C.R.\ acknowledges financial support by the Heisenberg program of the
Deutsche Forschungsgemeinschaft.

\section{Finite gap potentials}
In this section, we will briefly review the construction and
some results on
finite gap potentials.
We will more
or less follow the representation given in
\cite{McK}. For further information
on this many-faceted
topic (for example, the connections to equations of the
KdV hierarchy),
see \cite{GRT,GesWei}. The needed facts from the
theory of compact Riemann surfaces can be found in \cite{FarKra,Spr}.

Let energies $E_0<E_1<\cdots < E_{2g}$ be given. The aim is to
construct a family of (quasi-periodic) potentials $V\in C^{\infty}
(\mathbb R)$ so that
the corresponding
operators $H=-d^2/dx^2+V(x)$
on $L_2(\mathbb R)$ have purely absolutely
continuous spectrum with precisely the prescribed gaps:
\begin{equation}
\label{spectrum}
\sigma(H)= \mathbb R \setminus \bigcup_{n=0}^g
(E_{2n-1},E_{2n})\quad\quad\quad (E_{-1}:= -\infty ).
\end{equation}
To this end, consider the
Riemann surface $S$ of
\[
R(z) = \left( \prod_{n=0}^{2g} (E_n -z) \right)^{1/2}.
\]
$S$ is compact and hyperelliptic and its genus is equal to $g$.
A topological model of $S$ can be obtained by gluing together two copies
of the extended complex plane cut along the gaps $(\infty,E_0),
(E_1,E_2),\ldots ,(E_{2g-1},E_{2g})$.

The points of $S$ may thus be viewed as pairs $\widehat{z} =
(z,R(z))$. Here, $z\in\mathbb C \cup \{ \infty \}$ is the
canonical projection of $\widehat{z} \in S$ onto
$\mathbb C \cup \{ \infty \}$, and $R(z)$, which is of course
already determined by $z$ up to a sign, shows on which sheet
of the surface $\widehat{z}$ lies. Put differently,
the canonical projection
$\widehat{z}\mapsto z$ gives a two-sheeted branched covering
of the Riemann sphere $\mathbb C \cup \{ \infty \}$ by $S$;
the preimages of $\infty,E_0,\ldots, E_{2g}$ are branch points
of order one.

Standard coordinates $\zeta$ on
$S$ can be defined as follows. If $\widehat{z}$ is not a branch point,
put $\zeta= z$ in a neighborhood of $\widehat{z}$; near a finite
branch point $\widehat{E}_j$, use $\zeta=(E_j-z)^{1/2}$, the sign being
determined by the sign of $R$, and near infinity, use similarly
$\zeta=(-z)^{-1/2}$.

There are precisely
$g$ linearly independent holomorphic differentials (also known
as Abelian differentials of the first kind) on $S$. One can
obtain unique basis elements by prescribing certain periods. We will
work with the standard normalization which amounts to demanding that
for $i,j=1,\ldots,g$,
\begin{equation}
\label{normal}
2 \int_{E_{2i-1}}^{E_{2i}} \omega_j = \delta_{ij}.
\end{equation}
(The notation is a little sloppy: The path of integration projects onto
$[E_{2i-1},E_{2i}]$, and $(-1)^i R\le 0$ on this path.) Note
that the left-hand side is just the integral of $\omega_j$ over the
cycle $a_i$ of a standard homology basis (see, e.g.,
\cite[p.\ 109ff]{GesWei}).
One can then show that the $\omega_j$ thus defined are of the form
\begin{equation}
\label{omega}
\omega_j = \frac{p_j(z)}{R(z)}\, dz,\quad
p_j(z) = c_j\prod_{i\not= j} (\lambda_i^{(j)}-z),
\end{equation}
with $\lambda_i^{(j)}\in (E_{2i-1},E_{2i})$, $c_j>0$. Of course,
this representation refers to the coordinate maps $\widehat{z}
\mapsto z$ discussed above.

The Abel-Jacobi map $\alpha$ sends positive divisors of degree $g$ (that is,
unordered collections of $g$ points from $S$) to the Jacobi variety of $S$,
which is the complex torus equal to $\mathbb C^g$ modulo the
period lattice of the holomorphic differentials. This map is
onto; in other words, the Jacobi inversion problem can be solved.
We will need the Abel-Jacobi map only for divisors of the
form $(\widehat{\mu}_1,\ldots,\widehat{\mu}_g)$
with $\mu_i\in [E_{2i-1},E_{2i}]$.
The Abel-Jacobi map is then given by
\begin{equation}
\label{ajm}
\alpha_i(\widehat{\mu}_1,\ldots,\widehat{\mu}_g)
= 2\pi \sum_{j=1}^g \int_{\widehat{E}_{2j-1}}^{\widehat{\mu}_j} \omega_i
\mod 2\pi;
\end{equation}
here, we take paths of integration whose projections lie entirely in
the corresponding gaps $[E_{2j-1},E_{2j}]$.
It follows from classical theorems of Abel and Jacobi
\cite[Chapter 10]{Spr} that $\alpha=(\alpha_1,\ldots,\alpha_g)$ is
a bijection from the set of divisors
specified above onto the real part of
the Jacobi variety $\mathbb T^g = [0,2\pi)^g$.
Alternately, this fact may be verified directly, using a representation
of the Abel-Jacobi map that will be derived below (see eq.\
\eqref{abeljac}).

Actually, \eqref{ajm} differs from the standard definition
of the Abel-Jacobi map by an
additive constant vector and the factor $2\pi$; the choice \eqref{ajm} is
more convenient here.

Now the stage has been set for the actual construction of the finite
gap potentials. Consider the following linear flow on $\mathbb T^g$:
$\phi_x \alpha_0= \alpha_0+ \nu x$, where the frequency vector
$\nu$ is given by
\[
\nu_j = 2\pi \text{ res} \left. \left( (-z)^{1/2}\omega_j \right)
\right|_{z=\infty}.
\]
Using the coordinate
$\zeta= (-z)^{-1/2}$ at $z=\infty$, we can easily evaluate
the residue to obtain $\nu_j=4\pi c_j$, where $c_j$
is the normalization constant of the polynomial $p_j$ (see
\eqref{omega}). Now pull back the flow $\phi_x$ to
the set of divisors $(\widehat{\mu}_1,\ldots,\widehat{\mu}_g)$,
using the Abel-Jacobi
map. In other words, define the functions $\widehat{\mu}_j(x)\in S$
($\mu_j(x)\in [E_{2j-1},E_{2j}]$) by requiring that
\[
\alpha(\widehat{\mu}_1(x),\ldots,\widehat{\mu}_g(x)) =
\alpha_0 + \nu x .
\]
Next, introduce a potential $V$ by the following trace formula:
\begin{equation}
\label{trace}
V_{\alpha_0}(x) = E_0 + \sum_{n=1}^g (E_{2n-1}+E_{2n}-2\mu_n(x)).
\end{equation}
One can then show that this family of potentials solves the inverse
problem stated at the beginning of this section. Namely, the
operators $H= -d^2/dx^2 +V_{\alpha_0}(x)$ on $L_2(\mathbb R)$ have
purely absolutely continuous spectrum equal to the set given
in \eqref{spectrum}. This follows from the following
representation of the diagonal of the Green function of $H$:
\begin{equation}
\label{Green}
G(x,x;z) = \frac{1}{2R(z)} \prod_{n=1}^g \left( \mu_n(x)-z\right) .
\end{equation}
This important formula is derived with the aid of the so-called
Baker-Akhieser function, which gives explicit expressions for
the solutions $y$ of the DE $Hy=zy$. See \cite{McK} for the
details.

Actually, the right-hand side of
\eqref{Green} defines a meromorphic function on $S$ (with
simple poles precisely at the finite branch points) for
every fixed $x\in\mathbb R$. The Green function, however, depends
on $z\in\mathbb C$; therefore, we must complement \eqref{Green}
by recalling that
$\text{Im }G(z) \text{ Im }z >0$ for $\text{Im }z\not= 0$.

It is useful (and probably also more natural) to interpret the
above recipe in a slightly different way. Namely, define
$f:\mathbb T^g\to \mathbb R$ implicitly by the trace formula:
\begin{equation}
\label{defff}
f(\beta)=E_0 + \sum_{n=1}^g (E_{2n-1}+E_{2n}-2\mu_n),
\end{equation}
where $\alpha(\widehat{\mu}_1,\ldots,\widehat{\mu}_g)=\beta$. Then the
finite gap potential $V_{\alpha}(x)$ is obtained by evaluating
$f$ along the trajectory of $\alpha$ under the flow
$\phi_x \alpha= \alpha+ \nu x$.

So, if $\nu$ is known, then $V_{\alpha}$ is computed
by inverting the Abel-Jacobi map. We remark parenthetically
that there is an ``explicit''
solution to this problem which uses Riemann theta functions
\cite{GRT,GesWei,Mum}, but these formulae are not of much use here.
\section{Pointwise estimates on finite gap potentials}
Since $\mu_j\in [E_{2j-1},E_{2j}]$, the trace formula \eqref{trace}
immediately implies the following bound on $V_{\alpha}(x)$:
\begin{equation}
\label{l1bd}
\sup_{x\in\mathbb R}
\left|V_{\alpha}(x)-E_0\right| \le \sum_{n=1}^g (E_{2n}-E_{2n-1}).
\end{equation}
In other words, $\|V_{\alpha}-E_0\|_{\infty}$
is bounded by the $\ell_1$-norm of the
sequence of the gap lengths $E_{2n}-E_{2n-1}$. It is also obvious
that nothing more can be said in general: Indeed, if the components
of the frequency vector $\nu$ are rationally independent, every
trajectory $\{ \phi_x\alpha: x\in \mathbb R \}$
is dense in the torus $\mathbb T^g$, and thus the
$L_{\infty}$-norm of $V_{\alpha}(x)-E_0$ is equal to the maximum
of $f-E_0$ over the torus, so \eqref{l1bd} holds with equality in
this case.
However, there still is hope that \eqref{l1bd} {\it can} be improved
if the supremum is only taken over a bounded (but large) interval
$0\le x\le L$. Then the problem is to choose
a trajectory whose initial piece
avoids those points of the torus where $|f|$ is large.
Our next major goal is to confirm this hope. This will occupy
us for the following four sections.

It will be convenient to
use the centers and the half-lengths of the gaps as new parameters.
So, define
\[
m_n = \frac{E_{2n-1}+E_{2n}}{2},\quad l_n=\frac{E_{2n}-E_{2n-1}}{2}.
\]
The finite gap potentials that are needed in the construction
underlying Theorem \ref{T1.1} have gaps which are small compared
to the bands $[E_{2n},E_{2n+1}]$. Therefore, we from now on
concentrate on this situation. To make this condition precise,
we introduce
\[
l=\max_{n=1,\ldots,g} l_n, \quad d=\min_{n=2,\ldots,g} (m_n-m_{n-1});
\]
our condition on the parameters of the construction will be
that $(l/d)\ln g$ is sufficiently small.

The following theorem is our main result on finite gap potentials.
It says that the
family of finite gap potentials $\{ V_{\alpha}: \alpha\in\mathbb T^g \}$
contains functions which are over long intervals
``almost'' bounded by the $\ell_2$-norm of the gap lengths (rather
than the $\ell_1$-norm).

According to the above remarks, we now use
the following parameters to describe finite gap potentials:
$g\in\mathbb N$ ($g\ge 2)$ is
the number of gaps; $E_0<m_1<\cdots < m_g$ and $l_1,\ldots, l_g >0$
describe the locations and the lengths of the gaps, respectively.
We require that the gaps do not touch or overlap. Clearly, this amounts
to demanding that $E_0<m_1-l_1$ and $m_n+l_n<m_{n+1}-l_{n+1}$ for
$n=1,\ldots, g-1$.
\begin{Theorem}
\label{T3.2}
Let $C_1,C_2>0$ be constants so that $C_1\le m_1-E_0$,
$m_g-E_0\le C_2$, and let $N\in \mathbb N_0$. Then there
exists a constant $C$, depending only on $C_1,C_2$, and $N$
(but not on the parameters of the finite gap potentials),
such that the following holds.
For every $L\ge 1$,
there exists an $\alpha\in
\mathbb T^g$ so that
\[
\sup_{0\le x\le L}\left|f(\alpha+\nu x)-\widehat{f}_0\right|
\le C \left[ g^{1/2}l \left(\ln (gL)\right)^{1/2}
+ gl (ld^{-1}\ln g)^{N+1} \right] ,
\]
where $f$ was defined in \eqref{defff} and
\[
\widehat{f}_0 = \int_{\mathbb T^g} f(\beta) \,
\frac{d\beta}{(2\pi)^g}.
\]
\end{Theorem}
Recall from Sect.\ 2 that $f(\alpha +\nu x)$ is just the
finite gap potential $V_{\alpha}(x)$. In the proof of
Theorem \ref{T3.2}, we will in fact show that the
assertion holds with large probability if $\alpha\in
\mathbb T^g$ is chosen at random.

The first term of the bound is the $\ell_2$-norm of the
gap lengths (as promised), times a logarithmic factor.
Of course, the point is that the increase in $L$ is slow,
so we can still take a relatively large $L$. Note, however, that
we no longer get an improvement over the trivial bound
$gl$ if $L$ is of the order $e^g$.
In the application of Theorem \ref{T3.2} in this paper,
we will have
$L \le g^{\gamma}$, and then Theorem \ref{T3.2} indeed gives
a good bound.

From a theoretical point of view,
a particularly neat situation arises when the
flow $\phi_x$ and thus also the finite gap potentials are
periodic with period $p$. In that case, one can take $L=p$ to
obtain a bound which is valid for all $x\in\mathbb R$.
This remark is not as academic as it may seem, because
one can show, using topological arguments, that in situations
with small gaps one can
get a periodic $\phi_x$ by slightly moving the centers
$m_n$. The period will be of the order $p\approx d^{-1}$.
See also \cite[Appendix C.2]{DKV} for statements of this type.

The second term of the above bound contains the $\ell_1$-norm $gl$, but
multiplied by an arbitrarily high power of
$ld^{-1} \ln g$. So Theorem \ref{T3.2} is interesting only
if this combination is small, but this is the case in our construction
for proving Theorem \ref{T1.1}.
What exactly ``small''
means obviously depends on $C$ and thus on $C_1,C_2,N$, but
on nothing else. This will be very important in the proof of
Theorem \ref{T1.1}, where we will apply Theorem \ref{T3.2} to
a whole sequence of finite gap potentials.

We would like to emphasize the fact that we do not
subtract $E_0$
from $V_{\alpha}(x)$ (which is perhaps the constant that comes
to mind first), but rather
the average of $f$ over the real part of the Jacobi variety.
This may be viewed as a renormalization, due to higher order terms.
Indeed, $E_0$ is the limiting value of $f$ at $l=0$; now the Theorem
says that the zeroth Fourier coefficient $\widehat{f_0}$ (which contains
also terms which are of higher order in the small parameter $l/d$)
gives a better constant approximation to $V_{\alpha}(x)$. This remark
is actually true for approximations by trigonometric polynomials
of arbitrarily high degree; we will comment on
this point again after having discussed the proof of Theorem \ref{T3.2}.

We will give this proof in the following
three sections. This is the plan of attack: We will first solve
the Jacobi inversion problem up to order $N$ in the small
parameter $ld^{-1}\ln g$. (As explained above, the problem of
computing finite gap potentials basically is the
Jacobi inversion problem, that is, the problem of inverting
the Abel-Jacobi map.) This will be done by expanding in
Fourier and Taylor series and solving the equations by
iteration. The expressions obtained in this way rapidly
get out of hand as $N$ increases. However, things become
surprisingly transparent if a graphical representation of the
perturbation series is introduced. This will be developed
in Sect.\ 5, after having discussed some preparatory
material in Sect.\ 4.

Then, in Sect.\ 6,
we extend classical methods, due to Salem and
Zygmund \cite{SZ}, for bounding random trigonometric
polynomials to finish the proof.
In its original version, this argument shows, for example,
that for a random choice of signs, $p(x)=\sum_{n=1}^N \pm a_n
\cos nx$ is almost bounded by the $\ell_2$-norm of its
coefficients: $\|p\|_{\infty} \le C \|a\|_2 (\ln N)^{1/2}$.
Theorem \ref{T3.2} can perhaps be viewed as a nonlinear
version of this result.

\section{Proof of Theorem \ref{T3.2}: Basic estimates}
We want to analyze the Abel-Jacobi map \eqref{ajm}.
We can parametrize the divisors $(\widehat{\mu}_1,\ldots,
\widehat{\mu}_g)$ (as usual, $\mu_j\in [E_{2j-1},E_{2j}]$) by
the points $(\psi_1,\ldots,\psi_g)$
of another copy of the torus $\mathbb T^g = [0,2\pi)^g$ (not to
be confused with the real part of the Jacobi variety) as follows:
Write
\begin{align}
\label{subst1}
\mu_j & = m_j - l_j \cos\psi_j, \\
\label{subst2}
R(\mu_j) & = R_j(\mu_j) i l_j \sin\psi_j,
\end{align}
where $R_j(z)= R(z)/\sqrt{(E_{2j-1}-z)(E_{2j}-z)}$. This definition
is not yet complete since the
sign of $R_j(\mu_j)$ on the right-hand side of \eqref{subst2}
also needs to be specified. Note that
$iR_j(\mu)$ is real and non-zero for $E_{2j-1}<\mu < E_{2j}$.
Therefore, it makes sense to require that
$iR_j(\mu)$ be positive for odd $j$ and
negative for even $j$ and $\mu$ as above.
So, for a given $\psi_j\in [0,2\pi)$, eq.\ \eqref{subst1} tells
us what the projection $\mu_j$ of $\widehat{\mu}_j$ is, while
\eqref{subst2} determines the sheet on which $\widehat{\mu}_j$ lies.
In particular, if $\psi_j + \psi'_j = 2 \pi$, then the corresponding
points $\widehat{\mu}_j, \widehat{\mu}'_j \in S$ have the
same projections but lie on different sheets.

The substitution \eqref{subst1}, \eqref{subst2} allows us to
write integrals involving the $\omega_j$ in a particularly
convenient way. Indeed, recalling \eqref{omega},
we see that the normalization condition
\eqref{normal} now takes the form
\begin{equation}
\label{normal'}
2 \int_0^{\pi} \left. \frac{p_j(\mu)}{iR_n(\mu)}
\right|_{\mu=m_n-l_n\cos\psi} \,
d\psi = \delta_{jn} .
\end{equation}
Similarly, the Abel-Jacobi map, now viewed as a map from
$\mathbb T^g$ to $\mathbb T^g$ (but still denoted by $\alpha$),
can be written as
\begin{equation}
\label{abeljac}
\alpha_j(\psi_1,\ldots,\psi_g) =
2\pi \sum_{n=1}^g \int_0^{\psi_n} \left. \frac{p_j(\mu)}{iR_n(\mu)}
\right|_{\mu=m_n-l_n\cos\psi} \, d\psi.
\end{equation}

Finally, the function $f$ from the trace formula for $V$ (see
\eqref{defff}) takes the following form
when expressed in terms of the new variables:
\begin{equation}
\label{f}
f=E_0+2\sum_{n=1}^g l_n \cos\psi_n.
\end{equation}

As already discussed, Theorem \ref{T3.2} is vacuous if
$ld^{-1} \ln g$ is not small (if $ld^{-1} \ln g \ge \epsilon$,
just take $C=4\epsilon^{-N-1}$). Thus only the case where
\begin{equation}
\label{epsilon}
ld^{-1} \ln g < \epsilon
\end{equation}
needs proof; here, $\epsilon>0$
can be chosen according to our needs and may depend on
$C_1,C_2$, and $N$ (but on nothing else). In addition to
the hypotheses of Theorem \ref{T3.2}, we will therefore assume
\eqref{epsilon} with a sufficiently small $\epsilon$
from now on.
In particular, the reader should keep in mind that \eqref{epsilon}
with a suitable $\epsilon=\epsilon(C_1,C_2,N)$
as well as the hypotheses of Theorem \ref{T3.2} are (tacit) assumptions
in all lemmas of Sect.\ 4--6.

{\it Notational remark.} In the sequel, we will use the
following conventions.
A ``constant'' (usually denoted by $C$)
is a number that only depends on
$C_1,C_2$, and $N$. In particular, the constants which are
implicit in the Landau notation $O(\cdots)$ may only depend on $C_1$,
$C_2$, and $N$.
We will sometimes write $a\lesssim b$ instead of
$a\le Cb$ (or $a=O(b)$);
here, $C$ is a constant in the sense just explained.
Similarly, $a\approx b$ is short-hand for two-sided estimates.
Finally, the value of $C$ may change from one formula to the
next, so there is nothing wrong with an inequality like
$C+1 \le C$ (to give a blatant example).

Assuming \eqref{epsilon},
we can analyze \eqref{normal'}, \eqref{abeljac} in some detail
by using Taylor expansions. The following lemma will get us started.
\begin{Lemma}
\label{L3.1}
For all $j,n=1,\ldots,g$,
the function $p_j(z)/R_n(z)$ is holomorphic
in a neighborhood of $[E_{2n-1},E_{2n}]$, and for all $s\in\mathbb N_0$,
\[
\max_{z\in[E_{2n-1},E_{2n}]} \left| \frac{d^s}{dz^s}
\frac{p_j(z)}{R_n(z)} \right | \le \left(\frac{C}{d}\right)^s
 \, \frac{Cs!}{d^{-1}|m_j-m_n|+1} .
\]
Moreover,
\[
c_j = \frac{(m_j-E_0)^{1/2}}{2\pi} \left( 1+
O((l/d)^2\ln g) \right).
\]
\end{Lemma}
{\it Proof.} The first assertion is obvious; in fact, it holds
on any simply connected
neighborhood of $[E_{2n-1},E_{2n}]$ that avoids the other
branch points. Thus, for $z\in [E_{2n-1},E_{2n}]$, we can use
the Cauchy formula to represent the derivatives:
\begin{equation}
\label{cauchy}
f^{(s)}(z) = \frac{s!}{2\pi i} \int_K
\frac{f(\zeta)}{(\zeta-z)^{s+1}}\, d\zeta .
\end{equation}
Here, we integrate over the contour $K=\{ \zeta=m_n+(d/2)e^{i\varphi}:
0\le\varphi\le 2\pi \}$ in counter-clockwise direction. Note
that $K$ is well separated
from all gaps $[E_{2i-1},E_{2i}]$.
In particular,
if $l/d$ is sufficiently small (for instance, $l/d \le 1/4$ will do),
then $|\zeta-z| \gtrsim d$ for all $\zeta\in K$,
$z\in[E_{2n-1},E_{2n}]$, so \eqref{cauchy} implies that
\begin{equation}
\label{taylor}
\max_{z\in[E_{2n-1},E_{2n}]} \left| f^{(s)}(z)
\right| \le  C(C/d)^s s! \, \max_{\zeta\in K} \left| f(\zeta) \right| .
\end{equation}
We want to apply this to $f=p_j/R_n$, so we need
to estimate $p_j$ and $R_n$: We have that for $\zeta\in K$,
\begin{align*}
\left| R_n(\zeta)\right| & = |\zeta-E_0|^{1/2}
\left(\prod_{i\not=n} |m_i-l_i-\zeta|\, |m_i+l_i-\zeta|
\right)^{1/2}\\
& \gtrsim
\prod_{i\not=n} |m_i-\zeta| \left( 1+
O\left( \frac{l^2}{(m_i-m_n)^2} \right) \right) .
\end{align*}
Here we used the fact that $m_n-E_0 \approx 1$ by the hypotheses
of Theorem \ref{T3.2}.

Similarly, since the unknown zeros $\lambda_i^{(j)}$ of $p_j$
satisfy $\lambda_i^{(j)}\in (E_{2i-1},E_{2i})$ and since
$c_j>0$, we obtain
\[
\left| p_j(\zeta)\right|  = c_j \prod_{i\not=j}
\left| \lambda_i^{(j)} -\zeta \right|
= c_j \prod_{i\not=j} \left| m_i - \zeta \right| \left( 1+ O\left(
\frac{l}{|m_i-m_n|+d} \right) \right).
\]
Now $|m_i-m_j|\ge d|i-j|$, so taking
logarithms, we see that for small $ld^{-1}\ln g$,
\begin{align*}
\prod_{i\not=j} \left( 1+O\left( \frac{l}{|m_i-m_n|+d} \right) \right)
& = 1+O(ld^{-1}\ln g),\\
\prod_{i\not=n} \left( 1+O\left( \frac{l^2}{(m_i-m_n)^2} \right) \right)
& = 1 + O((l/d)^2).
\end{align*}
Estimates of this type will be used quite often in the sequel.
Combining the bounds just proved, we get
\[
\left|\frac{p_j}{R_n}(\zeta)\right| \le
c_j\, \frac{C}{d^{-1}|m_j-m_n|+1},
\]
and the claim on the derivatives of $p_j/R_n$ would
follow with \eqref{taylor} if we knew already the asserted
formula for the $c_j$'s.

So, it only remains to prove the estimate on $c_j$ stated in
Lemma \ref{L3.1}.
Taylor's theorem with remainder gives
\begin{multline*}
\left. \frac{p_j(\mu)}{R_n(\mu)}\right|_{\mu=m_n-l_n\cos\psi}=
\frac{p_j(m_n)}{R_n(m_n)}-
\frac{d}{dz} \left(\frac{p_j}{R_n}\right)(m_n)\,
\, l_n\cos\psi +\\
 O\left( c_j\,
\frac{(l/d)^2}{d^{-1}|m_j-m_n|+1}\right).
\end{multline*}
Plug this into \eqref{normal'}. The first order term integrates
to zero. Also,
\[
R_n(m_n) = -i \sqrt{m_n-E_0} (1+O((l/d)^2))
\prod_{i\not=n} (m_i-m_n),
\]
so we obtain
\begin{multline}
\label{1.1}
\frac{2\pi c_j}{\sqrt{m_n-E_0}}
\frac{\prod_{i\not=j} (\lambda_i^{(j)}-m_n)}{\prod_{i\not=n}
(m_i-m_n)}(1+O((l/d)^2)) + \\
O\left( c_j\,
\frac{(l/d)^2}{d^{-1}|m_j-m_n|+1}\right) = \delta_{jn}.
\end{multline}
For $j\not=n$, \eqref{1.1} leads to
\[
\frac{\lambda_n^{(j)}-m_n}{m_j-m_n}(1+O(ld^{-1}\ln g)) =
O\left( \frac{l^2/d}{|m_j-m_n|} \right),
\]
thus $\lambda_n^{(j)}=m_n+O(l^2/d)$. Using this in \eqref{1.1}
with $j=n$, we finally obtain
\[
\frac{2\pi c_n}{\sqrt{m_n-E_0}} \prod_{i\not=n}
\left( 1+O\left( \frac{l^2d^{-1}}{|m_i-m_n|}\right)\right) = 1
+ O\left( c_n\, (l/d)^2\right),
\]
and the lemma follows. $\square$

We now expand the integrands of the Abel-Jacobi
map \eqref{abeljac} in a Fourier series. This, and not a
Taylor series, is the appropriate choice here, because it
gives the correct ``renormalized'' constant term immediately,
without contributions from higher order terms.
So write
\begin{equation}
\label{fourier}
\left.
\frac{2\pi p_j(\mu)}{iR_n(\mu)}\right|_{\mu=m_n-l_n\cos\psi}
= \sum_{m\in\mathbb Z} a_m(j,n) e^{im\psi}.
\end{equation}
Since the left-hand side is in
$C^{\infty}(\mathbb T)$ as a function of $\psi$,
this expansion converges uniformly. Moreover,
\begin{equation}
\label{amjn}
a_m(j,n)= \int_0^{2\pi}\left.
\frac{p_j(\mu)}{iR_n(\mu)}\right|_{\mu=m_n-l_n\cos\psi}
e^{-im\psi}\, d\psi,
\end{equation}
and, as a consequence, $a_0(j,n)=\delta_{jn}$ (by \eqref{normal'}).
\begin{Lemma}
\label{L4.1}
\[
\left| a_m(j,n)\right| \le
\frac{(Cl/d)^{|m|}}{d^{-1}|m_j-m_n|+1}
\]
\end{Lemma}
{\it Proof.}
This is trivially satisfied if $m=0$, so we suppose that
$m\not= 0$. Then, by Taylor's theorem and Lemma \ref{L3.1},
\[
\frac{p_j}{R_n}(m_n-l_n\cos\psi) =
\sum_{k=0}^{|m|-1} b_k(j,n) (-l_n\cos\psi)^k + \rho_{|m|}(\psi),
\]
where the remainder satisfies the estimate
\[
\left| \rho_{|m|}(\psi)
\right| \le \frac{(Cl/d)^{|m|}}{d^{-1}|m_j-m_n|+1}.
\]
Since $\int_0^{2\pi} \cos^k \psi\, e^{-im\psi}
\, d\psi =0$ for $|m|>k$, the
claim now follows from \eqref{amjn}.
$\square$

Using $a_0(j,n)=\delta_{jn}$,
we can now plug \eqref{fourier} into \eqref{abeljac} to write
the Abel-Jacobi map in the form
\begin{equation}
\label{4.3}
\alpha_j = \psi_j +
{\sum_{m\in\mathbb Z}}'
\sum_{n=1}^g \frac{a_m(j,n)}{im} (e^{im\psi_n}-1).
\end{equation}
Here and in the sequel, the prime at the sum sign indicates
omission of the term with $m=0$.
To obtain \eqref{4.3},
we have integrated \eqref{fourier} term by term, which is
allowed because of the uniform convergence.
We want to solve the system of equations \eqref{4.3}
for $\psi_1,\ldots,\psi_g$.
It is useful to separate the leading term, which, due
to the smallness of the $a_m(j,n)$'s expressed by Lemma \ref{L4.1}, is
$\alpha_j$. So, introduce $\theta_j$ by writing $\psi_j=
\alpha_j+\theta_j$; then \eqref{4.3} becomes
\begin{equation}
\label{1.4}
\theta_j + {\sum_{m}}' \sum_{n=1}^g \frac{a_m(j,n)}{im} (e^{im\alpha_n}
e^{im\theta_n}-1 ) = 0,
\end{equation}
and these equations must now be solved for the $\theta_j$'s. Actually,
we will compute the $\theta_j$'s
only up to an error of order $O((ld^{-1}\ln g)^{N+1})$. Note that by
Lemma \ref{L4.1} and \eqref{1.4},
\begin{equation}
\label{1.5}
|\theta_j| \le 2 {\sum_{m}}' (Cl/d)^{|m|} \sum_{n=1}^g
\frac{1}{d^{-1}|m_j-m_n|+1} \lesssim ld^{-1}\ln g,
\end{equation}
since $d^{-1}|m_j-m_n|\ge |j-n|$.
Now we keep only those terms of
\eqref{1.4} which are of order $\le N$ in the small parameter $ld^{-1}\ln g$,
and we iterate these new equations $N$ times. The following lemma
justifies this procedure; we get indeed a good approximation to
$\theta_j$.
\begin{Lemma}
\label{L4.2}
Define $\theta_j^{(0)}=0$,
\begin{align}
\theta_j^{(s+1)}=& -{\sum_{|m|\le N}}' \sum_{n=1}^g
\frac{a_m(j,n)}{im}\, ( e^{im\alpha_n} - 1) \nonumber\\
\label{rectheta}
& - {\sum_{|m|\le N}}' \sum_{n=1}^g
\frac{a_m(j,n)}{im}\, e^{im\alpha_n} \sum_{t=1}^{N-|m|}
\frac{ (im\theta_n^{(s)})^t}{t!},
\end{align}
$s=0,1,\ldots,N-1$. Then
$\left| \theta_j^{(N)} - \theta_j \right| \le
C (ld^{-1}\ln g)^{N+1}$.
\end{Lemma}
{\it Proof.} We will prove by induction that
\begin{equation}
\label{1.6}
\left| \theta_j^{(s)} - \theta_j \right| \le
C (ld^{-1}\ln g)^{s+1}
\end{equation}
for $s=0,1,\ldots,N$. For $s=0$, this is just \eqref{1.5}.
Now assume \eqref{1.6} holds for
some $s\ge 0$. We claim that then
\begin{equation}
\label{4.1}
\theta_{j}^{(s+1)} = -{\sum_{m\in\mathbb Z}}'
\sum_{n=1}^g \frac{a_m(j,n)}{im} \left(
e^{im\alpha_n} e^{im\theta_n^{(s)}}-1 \right)
+ O((ld^{-1}\ln g)^{N+1}).
\end{equation}
Indeed, comparison with \eqref{rectheta} shows that the error
from \eqref{4.1}, which we want
to bound by $C(ld^{-1}\ln g)^{N+1}$, is equal to
\begin{multline*}
{\sum_{|m|\le N}}' \sum_{n=1}^g \frac{a_m(j,n)}{im} \, e^{im\alpha_n}
\sum_{t=N+1-|m|}^{\infty} \frac{(im\theta_n^{(s)})^t}{t!} +\\
\sum_{|m|>N} \sum_{n=1}^g \frac{a_m(j,n)}{im} \left(
e^{im\alpha_n} e^{im\theta_n^{(s)}}-1 \right).
\end{multline*}
The induction hypothesis implies that
\[
\left| \theta_n^{(s)} \right| \le
\left| \theta_n^{(s)} - \theta_n \right|
+ \left| \theta_n \right| = O(ld^{-1}\ln g),
\]
so, by Lemma \ref{L4.1},
the first contribution to the error is bounded by a constant times
\begin{multline*}
{\sum_{|m|\le N}}' \sum_{n=1}^g
\frac{(l/d)^{|m|}}{d^{-1}|m_j-m_n|+1}\, (ld^{-1}\ln g)^{N+1-|m|}
\\ \lesssim
{\sum_{|m|\le N}}' (l/d)^{N+1}
(\ln g)^{N+2-|m|} \lesssim (ld^{-1}\ln g)^{N+1},
\end{multline*}
as desired.
Similarly, the second contribution to the error term can be
estimated by
\[
\sum_{|m|>N} \sum_{n=1}^g \frac{(Cl/d)^{|m|}}{d^{-1}|m_j-m_n|+1}
\lesssim \ln g \sum_{|m|>N} (Cl/d)^{|m|}
\lesssim (l/d)^{N+1} \ln g.
\]
This concludes the proof of \eqref{4.1}. Adding \eqref{4.1} and
\eqref{1.4}, we obtain
\[
\theta_j^{(s+1)} = \theta_j -{\sum_{m\in\mathbb Z}}'
\sum_{n=1}^g \frac{a_m(j,n)}{im}\, e^{im\alpha_n} \left(
e^{im\theta_n^{(s)}}-e^{im\theta_n} \right)
+ O((ld^{-1}\ln g)^{N+1}).
\]
Lemma \ref{L4.1} together with
\[
\left| e^{im\theta_n^{(s)}}-e^{im\theta_n} \right|
\lesssim |m| (ld^{-1}\ln g)^{s+1},
\]
which follows from the induction hypothesis, now yield the
induction statement \eqref{1.6} for $s+1$.
$\square$

\section{The Feynman rules}
We now introduce, as announced above,
a graphical representation of the terms obtained from
the recursion \eqref{rectheta}.
We have $\theta_j^{(0)}=0$
and
\[
\theta_j^{(1)}= -{\sum_{|m|\le N}}'
\sum_{n=1}^g \frac{a_m(j,n)}{im} (e^{im\alpha_n}-1).
\]
This latter expression can be represented by the following
graph:\\[0.5cm]
\setlength{\unitlength}{1cm}
\begin{picture}(6,1)
\put(1,0.5){\circle*{0.2}}
\put(1,0.5){\line(1,0){2.9}}
\put(1,0.5){\vector(1,0){1.6}}
\put(4,0.5){\circle{0.3}}
\put(0.8,0.8){$j$}
\put(4.2,0.8){$n$}
\put(2.4,0.2){$m$}
\end{picture}
\\[0.5cm]
Here is the recipe to recover $\theta_j^{(1)}$
from this graph: Associate the factor
$a_m(j,n)$ with the edge $m$ with vertices $j$ and $n$. The
circled vertex $n$ contributes a factor $e^{im\alpha_n}-1$,
where $m$ is the parameter of the incoming edge.
Finally, multiply by $i/m$ and sum
over $m=\pm 1,\ldots,\pm N$ and $n=1,\ldots, g$.

These rules, suitably generalized, also work for larger
values of $s$. At first sight, the formula for $\theta_j^{(2)}$
looks considerably more complicated than that for
$\theta_j^{(1)}$ because now the second line of \eqref{rectheta}
also contributes. However, it is not hard to convince oneself that
$\theta_j^{(2)}$ can actually be
computed by evaluating the following graphs.
\\[0.5cm]
\begin{picture}(10,6)
\put(1,5){\circle*{0.2}}
\put(1,5){\line(1,0){1.9}}
\put(1,5){\vector(1,0){1.2}}
\put(3,5){\circle{0.3}}
\put(0.8,5.2){$j$}
\put(3.1,5.2){$n$}
\put(1.9,4.6){$m$}

\put(3.9,5){$+$}

\put(5,5){\circle*{0.2}}
\put(5,5){\line(1,0){2}}
\put(5,5){\vector(1,0){1.1}}
\put(7,5){\circle*{0.2}}
\put(7,5){\line(1,0){1.9}}
\put(7,5){\vector(1,0){1.2}}
\put(9,5){\circle{0.3}}
\put(4.8,5.2){$j$}
\put(6.9,5.2){$n_1$}
\put(5.9,4.6){$m_1$}
\put(9.1,5.2){$n_2$}
\put(7.9,4.6){$m_2$}

\put(9.9,5){$+$}

\put(1,2){\circle*{0.2}}
\put(1,2){\line(1,0){2}}
\put(1,2){\vector(1,0){1}}
\put(3,2){\circle*{0.2}}
\put(0.8,2.2){$j$}
\put(1.9,1.6){$m_1$}
\put(2.9,2.2){$n_1$}
\put(3,2){\line(2,1){1.9}}
\put(3,2){\vector(2,1){1}}
\put(3,2){\line(2,-1){1.9}}
\put(3,2){\vector(2,-1){1}}
\put(5,3){\circle{0.3}}
\put(5,1){\circle{0.3}}
\put(4,2.2){$m_2$}
\put(4,1.1){$m_3$}
\put(5.1,3.2){$n_2$}
\put(5.1,0.6){$n_3$}

\put(6,2){$+\quad\quad \cdots$}
\end{picture}
\\[0.5cm]
More precisely, there are $N$ such graphs; they have the common
property that every edge except the first one emanates from
the second vertex.
Again, edges contribute factors of
the form $a_m(n,n')$, and for each vertex $\not=j$,
there is a factor $e^{im\alpha_n}$ ($e^{im\alpha_n}-1$ if
the vertex is marked by a circle).
Then one has to multiply by a factor that depends on the
edge indices $m_i$ and also on the graph
and finally sum over all parameters except $j$. (Explicitly,
this factor is
\[
i(-1)^{E+1}\frac{m_1^{E-2}}{(E-1)!} \prod_{k=2}^E m_k^{-1},
\]
where $E$ is the number of edges.)

We are now ready to formulate the rules for computing
$e^{i(\alpha_j+\theta_j)}$ from graphs of this type. The
quantity $e^{i(\alpha_j+\theta_j)}=e^{i\psi_j}$
is of especial interest here
because the function $f$ from \eqref{f} depends on exactly this
combination.\\
\begin{center}
{\it Feynman rules for $e^{i(\alpha_j+\theta_j)}$}
\end{center}
\vspace{0.2cm}
\begin{enumerate}
\item Draw all directed trees with at most $N$ edges. By a
``directed tree'', we mean a connected graph with the property that
there is precisely one vertex with only outgoing edges, while
for every other vertex, there is exactly one incoming edge.
The vertices without outgoing edges are called final vertices
(the trivial graph consisting of just one vertex is excluded
in this definition); they are marked by circles.

Formally, such a graph (with $E$ edges, say)
may be represented by $E+1$ symbols
$V_1,\ldots,V_{E+1}$ (``vertices'') and a collection of
$E$ ordered pairs $(V_i,V_j)$ with $i\not= j$ (``edges''). Two
graphs are equal if there is a bijection from one set of vertices
to the other which preserves the edges.

The figure below illustrates the case $N=3$.\\[0.5cm]
\setlength{\unitlength}{0.7cm}
\begin{picture}(15,3)
\put(1,3){\circle*{0.2}}

\put(3,3){\circle*{0.2}}
\put(3,3){\line(1,0){1.9}}
\put(3,3){\vector(1,0){1}}
\put(5,3){\circle{0.3}}

\put(6,3){\circle*{0.2}}
\put(6,3){\line(1,0){1.4}}
\put(6,3){\vector(1,0){0.8}}
\put(7.5,3){\circle*{0.2}}
\put(7.5,3){\line(1,0){1.4}}
\put(7.5,3){\vector(1,0){0.8}}
\put(9,3){\circle{0.3}}

\put(10,3){\circle*{0.2}}
\put(10,3){\line(1,0){1.4}}
\put(10,3){\vector(1,0){0.8}}
\put(11.5,3){\circle*{0.2}}
\put(11.5,3){\line(1,0){1.4}}
\put(11.5,3){\vector(1,0){0.8}}
\put(13,3){\circle*{0.2}}
\put(13,3){\line(1,0){1.4}}
\put(13,3){\vector(1,0){0.8}}
\put(14.5,3){\circle{0.3}}

\put(0,1){\circle*{0.2}}
\put(0,1){\line(2,1){1.9}}
\put(0,1){\line(2,-1){1.9}}
\put(0,1){\vector(2,1){1}}
\put(0,1){\vector(2,-1){1}}
\put(2,2){\circle{0.3}}
\put(2,0){\circle{0.3}}

\put(3,1){\circle*{0.2}}
\put(3,1){\line(2,1){1.9}}
\put(3,1){\line(2,-1){1.9}}
\put(3,1){\vector(2,1){1}}
\put(3,1){\vector(2,-1){1}}
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\vspace{0.4cm}
\item For every graph,
label the (unique) vertex without incoming edge $j$.
Then, attach the indices $n_1,\ldots, n_E$ to the remaining vertices,
and label the edges $m_1,\ldots , m_E$.
It is of no significance how the indices $n_1,\ldots, n_E$ and
$m_1,\ldots, m_E$ are assigned to the vertices and edges, respectively,
but once a graph
has been labeled, this particular labeling is fixed
once and for all.
\item These labeled graphs are translated into formulae as follows.
An edge labeled $m$ pointing from vertex $n$ to vertex $n'$
stands for a
factor $a_m(n,n')$. A non-final
vertex with index $n\not=j$ contributes
$e^{im\alpha_n}$, where $m$ is the (unique) incoming edge.
In case $n$ is a final vertex, the rule is similar except that
the factor now is
$e^{im\alpha_n}-1$. The vertex $j$ always carries the
factor $e^{i\alpha_j}$. Finally, the result is multiplied
by a number $c_G(m_1,\ldots,m_E)$ which depends on the graph and
the edge indices (and $N$, but
this is fixed throughout). In principle, $c_G$ can be computed,
as the discussion below will show, but we do not need to know
the precise values of the $c_G$'s here.
\item Sum over $m_i\not= 0$, $\sum_{i=1}^E |m_i|\le N$,
and $n_i=1,\ldots, g$. Finally, sum over all graphs.
\end{enumerate}

Carrying out these instructions produces a (complicated)
function of the $\alpha_n$'s. The claim is that up to an
error $O((ld^{-1} \ln g)^{N+1})$, this function coincides with
$e^{i(\alpha_j+\theta_j)}$. We will now prove
this assertion, which is the central result of this section.

This proof, though not really difficult, is not
easy to formulate; to get a feeling for the underlying
principles, it is advisable to try things out
by iterating \eqref{rectheta} a few times and
drawing some pictures. Our verbal description will thus be somewhat
sketchy.

The strategy of the proof, however, is straightforward. First of all,
we show that the approximations $\theta_j^{(s)}$ from \eqref{rectheta}
admit a representation by diagrams. We have demonstrated this
already for $s=1,2$, and the general case is hardly more difficult.
Then, we use this knowledge to formulate similar rules for
$e^{i(\alpha_j+\theta_j)}$. Finally, terms of order $(ld^{-1}\ln g)^{N+1}$
or higher can be dropped on the way.

So, our first claim is the following statement:
The $\theta_j^{(s)}$ from
Lemma \ref{L4.2} can be calculated by evaluating certain
graphs according to similar rules like the ones given above.
There are a number of differences: All graphs have exactly one edge emanating
from $j$, there is no factor $e^{i\alpha_j}$ attached to
$j$, the factors $c_G$ are different, the $m_i$'s are
summed over the range $|m_i|\le N$, $m_i\not= 0$, and there
may be graphs with more than $N$ edges.

In fact, we know this already for $s=1,2$, and the proof of
the general case is by induction on $s$. By its definition,
$\theta_j^{(s+1)}$
is obtained by inserting $\theta_n^{(s)}$ on the right-hand
side of \eqref{rectheta}. By induction hypothesis, $\theta_n^{(s)}$
is a sum of many terms each of which corresponds to a graph with
certain parameters. We now multiply out the right-hand side
of \eqref{rectheta} and only then take the various sums.
To prove our claim, it suffices to make
the following observations: graphs are multiplied together by
attaching them to one another at the ``initial'' vertex $j$.
Similarly, multiplying a graph by $a_m(j,n)e^{im\alpha_n}$, as
in the second line of \eqref{rectheta}, amounts to attaching
this graph to the single-edge graph
in such a way that the final vertex of this single-edge graph and
the initial vertex of the other graph combine to one new vertex.

Also, we may restrict ourselves to graphs with at most $N$
edges and to parameters $m_i$ with $\sum_{i=1}^E |m_i|\le N$.
Indeed, since each edge carries a factor $a_m(n,n')$, Lemma
\ref{L4.1} implies that the omitted contributions are
$O((ld^{-1}\ln g)^{N+1})$. Here, the logarithmic factors come
from the denominators of the bound of Lemma \ref{L4.1}, when the
vertex indices are summed over. Note also in this context that
summing over the $m_i$'s is never dangerous because the restrictions
$|m_i|\le N$ imply that there is an a priori bound (depending
on $N$ only) on the number
of summands.

Next, we have that
\[
e^{i(\alpha_j+\theta_j)} = e^{i\alpha_j}\sum_{t=0}^N
\frac{(i\tilde{\theta}_j^{(N)})^t}{t!} + O((ld^{-1}\ln g)^{N+1}).
\]
The tilde on the right-hand side indicates the omission
of higher order terms, as discussed in the preceding paragraph.
Again, the task is to multiply this out. The $\tilde{\theta}_j^{(N)}$
have graphical representations, as we have just seen, and the
above remarks about multiplying together different graphs are
still relevant here. The asserted rules follow from this. The
additional factor $e^{i\alpha_j}$ has simply been attached to
the vertex $j$.

Note also that the same graph may arise many times when the process
of multiplying out is performed, but then we can simply combine
these contributions to a single one. This will only affect the
numbers $c_G(m_1,\ldots,m_E)$.
\section{Bounds along a random trajectory}
This last part of the proof of Theorem \ref{T3.2} deals
with the problem of bounding $f(\alpha)-\widehat{f}_0$ along
trajectories $\alpha=\alpha_0+\nu x$, given the information
obtained in the preceding section. First of all, recall from \eqref{f} that
\begin{equation}
\label{f0}
f(\alpha)= E_0+2\sum_{j=1}^g l_j\cos\psi_j
=E_0 +2 \text{ Re }\sum_{j=1}^g l_j e^{i(\alpha_j+\theta_j)}.
\end{equation}
We will first convince ourselves that $f$ is of the form
\begin{equation}
\label{ff}
f(\alpha)=\widehat{f}_0 + \sum\nolimits_{|m|_1\le N+1}'
b(m)\sin (m\cdot\alpha+\varphi_m)
+O(lg(ld^{-1}\ln g)^{N+1}).
\end{equation}
We use a slightly different notation in this section in that
now $m=(m_1,\ldots,m_g)$ with $m_i\in\mathbb Z$. Also,
$|m|_1=\sum |m_i|$ and $m\cdot\alpha = \sum m_i\alpha_i$;
finally, the prime at the sum sign now means omission
of the summand with $m=(0,\ldots,0)$.

To prove \eqref{ff}, use \eqref{f0} and think of the
exponentials $e^{i(\alpha_j+\theta_j)}$ as being evaluated
according to the Feynman rules. Then $\alpha$-dependent
factors come in only through the vertices of the graphs;
more precisely, vertices contribute factors of the form
$e^{im\alpha_n}$ (or $e^{im\alpha_n}-1$ if the vertex is final),
where $m$ is the index of the incoming
edge. The vertex $j$ always contributes a factor $e^{i\alpha_j}$,
so each graph is a sum of $\alpha$-independent factors
times an exponential of the form $\exp(i(\alpha_j
+\sum m_i\alpha_{n_i}))$. Since rule 4 imposes the
restriction $\sum |m_i| \le N$, a rearrangement of terms
gives \eqref{ff}, as asserted.

Clearly, this argument has not only established \eqref{ff},
but it has also indicated how the coefficients $b(m)$ can
be computed, at least in principle, using the graphs introduced
in Sect.\ 5. This will become very important in a moment.
(Just proving \eqref{ff} is easy and does not require the
Feynman rules.)

To prove Theorem \ref{T3.2}, we need to estimate the
second term on the right-hand side of \eqref{ff}. Call
this sum $f_N(\alpha)$. The main step will be to prove the
following estimate. Given Lemma \ref{L5.1}, we will then be
able to apply the methods of \cite{SZ}.
\begin{Lemma}
\label{L5.1}
There is a constant $C$, so that for every $\lambda\in\mathbb R$,
\[
\int_{\mathbb T^g} e^{\lambda f_N(\alpha)}\, \frac{d\alpha}{(2\pi)^g}
\le e^{C\lambda^2 l^2g}.
\]
\end{Lemma}
{\it Remark.} Our ``definition'' of $f_N$ is not quite complete,
since \eqref{ff} does {\it not} uniquely determine $f_N$, given
$f$. Lemma \ref{L5.1} really asserts that for some fixed choice of
$f_N$, consistent with \eqref{ff}, the stated estimate holds.
More precisely, $f_N$ is obtained by going from \eqref{f0} to
\eqref{ff} in exactly the way described above.
The following proof will also clarify this.

{\it Proof.}
We will further decompose $f_N$ and then analyze the individual
terms separately. To this end, we first introduce equivalence classes
of indices $m$. Namely, we say that $m$ and $m'$ are
equivalent if they have the same non-zero entries, taking
the order into account. To put
this into more formal language, write
\[
m=(0,\ldots,0,k_1,0,\ldots,0,k_2,0,\ldots,0,k_r,0,\ldots,0),
\]
with $r\in\mathbb N$ and $k_i\not= 0$ for all $i=1,\ldots,r$.
Then $m$ and $m'$ are equivalent precisely if $r=r'$ and
$k_i=k'_i$ for all $i$.

This definition may not look very useful at first sight, but
recall that $N$ (which bounds the $\ell_1$-norm of $m$) is
fixed while $g$ (which is the length of the vectors $m$)
is typically large, so the vectors $m$ indeed have only relatively
few non-zero entries.

The number of equivalence classes in the set of indices
$\{ m\in\mathbb Z^g : |m|_1 \le N+1 \}$
only depends on $N$, but not
on $g$. (Note, however, that the cardinality of the equivalence
classes themselves does go to infinity as $g$ increases.)
Now fix an equivalence class $(m_0)$ and consider
\[
\sum_{m\in(m_0)} b(m)\sin(m\cdot\alpha+\varphi_m).
\]
Denote the positions of the
non-zero entries $k_i$ of $m\in (m_0)$ by $n_i$. Then, if we vary the
$n_i$'s (respecting the obvious restrictions $1\le n_1<n_2<
\cdots < n_r \le g$), but keep $r$ and the $k_i$'s fixed,
we get exactly all elements of the equivalence
class under consideration. Thus the above sum is equal to
\begin{equation}
\label{5.1}
\sum_{1\le n_1<\cdots < n_r \le g} b(m)\sin\left(
k_1\alpha_{n_1}+\cdots + k_r\alpha_{n_r} + \varphi_m\right) .
\end{equation}
In this formula, $m$ is defined
by $m_{n_i}=k_i$ and $m_i=0$ otherwise.

Using the addition laws for sine and cosine $r-1$ times, we can
write \eqref{5.1} as a sum of $2^{r-1}$ terms of the form
\begin{equation}
\label{5.2}
\sum_{1\le n_1<\cdots < n_r \le g} b(m)\sin (k_1\alpha_{n_1}+\gamma_1)
\cdots \sin (k_r\alpha_{n_r}+\gamma_r),
\end{equation}
Here, the dependence of the phases $\gamma_i$ on the index $m$
has not been made explicit because the precise values of the
$\gamma_i$'s will not matter anyway. (On top of that,
we of course have a lot of
freedom in the choice of the $\gamma_i$'s, given $\varphi_m$ from
\eqref{5.1}.)

Since $r\le N+1$, we
still have a universal bound (depending on $N$ only)
on the number of different sums of the form
\eqref{5.2} that arise in the decomposition
of $f_N$ just performed.
Fix such a sum
and call it $F=F(\alpha_1,\ldots,\alpha_g)$ for easier
reference.

It suffices to establish the lemma with
$F$ in place of $f_N$. Indeed, if this is proved, then, since
$f_N=\sum F$ with an a priori bound on the number of summands
$F$, the claimed estimate follows from H\"older's
inequality.

Now, to bound $\int e^{\lambda F}$,
we first do the integration with respect to the
last variable $\alpha_g$. The sum defining $F$ contains many
terms that do not depend on $\alpha_g$. More precisely, we have that
\[
F(\alpha_1,\ldots,\alpha_g) = F_1(\alpha_1,\ldots,\alpha_{g-1})
\sin(k_r\alpha_g + \gamma_r) + F_2(\alpha_1,\ldots,\alpha_{g-1}),
\]
with
\begin{align*}
F_1 & =
\sum_{1\le n_1<\ldots<n_{r-1}\le g-1 \atop n_r=g}
b(m) \sin(k_1\alpha_{n_1}
+\gamma_1)\cdots  \sin(k_{r-1}\alpha_{n_{r-1}}+\gamma_{r-1}),\\
F_2 &  =
\sum_{1\le n_1<\ldots<n_r\le g-1} b(m) \sin(k_1\alpha_{n_1}
+\gamma_1)\cdots  \sin(k_r\alpha_{n_r}+\gamma_r).
\end{align*}

So, we now have to evaluate $(2\pi)^{-1} \int_0^{2\pi} \exp[\lambda F_1
\sin(k_r\alpha + \gamma_r)]\, d\alpha$. The substitution
$\beta=k_r\alpha + \gamma_r$ together with the computation
\[
\int_0^{2\pi} e^{c\sin\beta}\, \frac{d\beta}{2\pi}
=\sum_{n=0}^{\infty}\frac{c^n}{n!}\int_0^{2\pi}
\sin^n\beta\, \frac{d\beta}{2\pi}
=\sum_{n=0}^{\infty}\frac{(c/2)^{2n}}{(2n)!} \binom{2n}{n}
\le e^{c^2/4}
\]
show that
\begin{align}
\int_0^{2\pi} e^{\lambda F}\, \frac{d\alpha_g}{2\pi}& =
e^{\lambda F_2} \int_0^{2\pi}
e^{\lambda F_1 \sin(k_r\alpha_g+\gamma_r)}\, \frac{d\alpha_g}{2\pi}
\nonumber\\
\label{5.5}
& \le  e^{\lambda F_2} \exp\left( \frac{\lambda^2}{4}
\left(\sum
|b(m)| \right)^2 \right) ,
\end{align}
where the sum is over $1\le n_1<\ldots <n_{r-1}\le g-1$, and
$n_r=g$. We now estimate this sum $\sum |b(m)|$. By the
discussion following \eqref{ff}, the coefficients $b(m)$ can be
obtained with the help of the Feynman rules of the preceding section
by collecting those contributions which depend on $\alpha$
in exactly the way described by $m$. (More precisely, collect
everything that comes with a factor $e^{im\cdot\alpha}$, multiply
by the corresponding $l_j$'s, and then
take the real part and read off $b(m)$.) By the triangle inequality,
we can estimate $\sum |b(m)|$ by bounding the individual contributions
associated with certain fixed graphs with fixed labelings and then taking
the various sums at the very end.

So, fix a graph that contributes to some $b(m)$ occuring in
the sum $\sum |b(m)|$.
To avoid confusion with the numbers
$m_i,n_i$ introduced in this section, the parameters labeling
the edges and vertices of the graph will now be called
$m'_i$ and $n'_i$, respectively.
Since the factors $e^{im'\alpha_{n'}}$ are
attached to the vertices of the graph and since the $m$'s
under consideration have $r$ non-zero entries, the graph
fixed above must have at least $r$ vertices and hence at least
$E\ge r-1$ edges.
Moreover, since $n_r$ is set equal to $g$ in the sum
$\sum |b(m)|$ we are trying to estimate, at least one vertex
of the graph must have its parameter equal to $g$. In other
words, $j=g$ or $n'_i=g$ for some $i$. (There is the additional
restriction that $\sum |m'_i| \ge \sum_{i=1}^r |k_i| -1$, but
this will not be used.)

Armed with these observations, we are now ready to do the
estimates. By the Feynman rules, the contribution
coming from the fixed
graph admits a bound of the form
\begin{equation}
\label{5.6}
Cl \left|a_{m'_1}(\ldots) \cdots a_{m'_E}(\ldots) \right|
\end{equation}
if the parameters $j,n'_1,\ldots,n'_E,m'_1,\ldots,m'_E$ are
all kept fixed. The factor $l$ allows for the fact that
the $l_j$'s from \eqref{f0} have been absorbed by the
$b(m)$'s when passing to \eqref{ff}.
The arguments of the factors $a_{m'_i}$ depend
on the particular form of the graph and also on the
way the graph was labeled.
Note that the (unknown)
factors $c_G$ from rule 3 can be absorbed
by $C$ because there are only finitely many different values of
$c_G(m'_1,\ldots,m'_E)$ and
we can thus simply estimate these
numbers by their maximum.

We now use Lemma \ref{L4.1} to estimate \eqref{5.6}, and
we want to sum these bounds over those values of the
parameters $j,n'_i,m'_i$ which satisfy the restrictions
obtained above. (In contrast to the rules from Sect.\ 5, there
is now a sum over $j=1,\ldots, g$ also; this is simply the
sum from \eqref{f0}.) In particular, $j$ or one of the $n'_i$'s
is held fixed (equal to $g$). This implies that we can
sum over the indices of the remaining vertices in the following
way: First of all, delete the vertex corresponding to the fixed
index. From the remaining graph,
pick a vertex which is connected to just one edge
and perform the corresponding sum $n'_i=1,\ldots,g$.
By the choice of the vertex, $n'_i$ appears in precisely
one of the factors $a_{m'_i}$ as the argument. The denominator
of the bound of Lemma \ref{L4.1} thus yields a factor
$\lesssim \ln g$ when summed in the way just described.
As a reminder that the corresponding sum has been performed,
delete the chosen vertex together with the edge connected to it.
Then repeat the whole procedure with the modified graph to determine
the next index to be summed over. Again, just one $a_{m'_i}$ is involved
in the sum, and thus another factor $\lesssim \ln g$ results.
Since at each stage, there are equally many vertices and edges,
this process can only stop after the whole graph has disappeared.
There are only $E+1\le N+1$ possible choices for the vertex
whose parameter is set equal to $g$,
so the net result is that after summation
over the vertices, \eqref{5.6} can be estimated by
\begin{equation}
\label{5.3}
C l(l/d)^{|m'_1|+\cdots +|m'_E|}(\ln g)^E.
\end{equation}
Indeed, the numerators
from Lemma \ref{L4.1} contribute the factor $(l/d)^{|m'_1|+\cdots
+|m'_E|}$, and
by the argument just given,
each of the $E$ sums
over the vertices accounts for a factor $\ln g$. 

We can further estimate \eqref{5.3}. By rule 4, $m'_i\not=0$ for all
$i=1,\ldots,E$, hence $\sum |m'_i|\ge E$; since
$E\ge 0$, we can thus
can bound \eqref{5.3} simply by $Cl$.

Now the rest is easy. First of all, each of the at most
$N$ parameters $m'_i$
has values in $\pm 1,\ldots,\pm N$, so summing over the
$m'_i$'s just increases the constant $C$ (by at most a
factor $(2N)^N$). Then, the total number of graphs also depends
on $N$ only, so we can finally sum over those graphs which could
in principle contribute to $\sum |b(m)|$, and we still have
the bound $Cl$ (again, with a possibly larger constant $C$).

Returning to \eqref{5.5} now, we have thus proved that
\[
\int_{\mathbb T^g} e^{\lambda F(\alpha)}\,
\frac{d\alpha}{(2\pi)^g} \le e^{C\lambda^2l^2}
\int_{\mathbb T^{g-1}} e^{\lambda F_2(\alpha')}\,
\frac{d\alpha'}{(2\pi)^{g-1}},
\]
where $\alpha=(\alpha_1,\ldots,\alpha_g)$ and
$\alpha'=(\alpha_1,\ldots,\alpha_{g-1})$. The integral on the
right-hand side has the same structure as the original one,
except that $g$ has been replaced by $g-1$. We can therefore
repeat the whole argument; the second step would be
to carry out the integration with respect to $\alpha_{g-1}$ in
the same way as discussed above. We need at most $g$ steps
to do the integral completely, and at each step, we get a
factor $\exp(C\lambda^2l^2)$.
As a result, we obtain the estimate $(2\pi)^{-g}\int e^{\lambda F} \le
e^{C\lambda^2l^2 g}$, and, as already explained, the lemma
follows. $\square$

{\it Remarks.} 1.\ The key point of this proof was the
observation that the summation over the vertices only gives
logarithmic factors $(\ln g)^E$, but no powers of $g$. Note that
to establish this, in turn, we only used some structural
information contained in the Feynman rules and Lemma \ref{L4.1};
the precise form of the underlying iteration was largely irrelevant.

2.\ The estimate $Cl$ on \eqref{5.3} is of course crude unless
we are in the extreme case $E=0$ (which, it turns out now, gives
the dominant contributions to $f_N$). Indeed, to prove the extension
of Theorem \ref{T3.2} mentioned in the beginning of the Introduction
and in Sect.\ 3, one has
to keep \eqref{5.3} as it stands.

We are now ready to finish the proof of Theorem \ref{T3.2}.
This final part of the argument uses methods
developed in \cite{SZ}. We will follow the
presentation given in \cite[Chapter 6]{Kah}.
We want to bound $f_N$ along a trajectory $\phi_x \alpha=
\alpha +\nu x$. So, let
\[
M(\alpha) = \max_{0\le x\le L} \left| f_N(\alpha + \nu x)\right| .
\]
To run the argument from \cite{Kah,SZ},
we need a bound on $(d/dx)f_N(\phi_x\alpha)$. Write $f_N$
as a sum of terms of the form \eqref{5.2},
as in the proof of Lemma \ref{L5.1}, and fix again
one of the summands $F$. Then,
by the argument presented in that proof, the sum over the
corresponding coefficients $b(m)$ satisfies
\[
\sum_{1\le n_1<\ldots<n_r\le g} |b(m)| \le Cgl
\]
Indeed, to show this, it just remains to sum the bound $Cl$ also
over the vertex index that had been fixed, and this gives an additional
factor $g$. Since
\[
\frac{d}{dx} F(\alpha+\nu x)  =
\sum b(m) \sum_{j=1}^r k_j\nu_{n_j}
\cos(k_j(\alpha_{n_j}+\nu_{n_j} x)+\gamma_j)
\prod\nolimits_j\: ,
\]
where $\prod_j$ is short-hand for the product of sines with the
$j$th factor omitted, it now follows that
\[
\left| \frac{d}{dx} F(\alpha+\nu x) \right| \le C \sum |b(m)|
\le C gl .
\]
Here we also used the fact the $\nu_n$'s are bounded; this
follows from Lemma \ref{L3.1} since $\nu_n=4\pi c_n$.
Summing the above bounds, we see that also
$\left| (d/dx)f_N(\phi_x\alpha)\right| \le Cgl$.

The maximum $M(\alpha)$ is attained at some point $x_0$, and thus
by the mean value theorem, there is a constant $C_0$ (as usual,
depending only on $C_1$, $C_2$, and $N$) together with
an interval $I=I(\alpha)
\subset [0,L]$
of length $|I|=\min \{ C_0M(\alpha)/(gl), L\}$,
so that $|f_N(\alpha+\nu x)| \ge M(\alpha)/2$ for all $x\in I$.
We may assume that $M(\alpha)\ge C_0^{-1}  l$
for all $\alpha\in \mathbb T^g$, since in the opposite case we
have for free a better bound than the one we are trying to prove.
We then have that $g |I| \ge 1$,
and it follows that
\begin{align*}
\int_{\mathbb T^g} e^{\lambda M(\alpha)/2}\,
\frac{d\alpha}{(2\pi)^g} & \le  g
\int_{\mathbb T^g} |I(\alpha)| e^{\lambda M(\alpha)/2}\,
\frac{d\alpha}{(2\pi)^g}\\
& \le g \int_{\mathbb T^g}\frac{d\alpha}{(2\pi)^g}
\int_{I(\alpha)} dx \left( e^{\lambda f_N(\alpha+\nu x)}
+ e^{-\lambda f_N(\alpha+\nu x)} \right) \\
& \le g \int_0^L dx\int_{\mathbb T^g}\frac{d\alpha}{(2\pi)^g}\,
\left( e^{\lambda f_N(\alpha+\nu x)}
+ e^{-\lambda f_N(\alpha+\nu x)} \right)\\
& = gL \int_{\mathbb T^g}\frac{d\alpha}{(2\pi)^g}\,
\left( e^{\lambda f_N(\alpha)}
+ e^{-\lambda f_N(\alpha)} \right)\\
& \le 2gL e^{C\lambda^2 l^2 g} .
\end{align*}
The last step is by Lemma \ref{L5.1}. For $\lambda>0$,
we can write this inequality
in the form
\[
E\left( \exp \left( \frac{\lambda}{2} \left[
M - 2C\lambda l^2 g - \frac{2}{\lambda}
\ln (4gL) \right] \right) \right) \le \frac{1}{2} ,
\]
with $E(\cdots)$ denoting the expectation taken with respect
to the probability measure $(2\pi)^{-g}d\alpha$ on the torus
$\mathbb T^g$. By a Chebyshev estimate, the inequality
\[
M(\alpha) \le 2C\lambda gl^2  + \frac{2}{\lambda}
\ln (4gL)
\]
holds with probability $\ge 1/2$. The parameter $\lambda>0$ is
still at our disposal, the optimal choice being
\[
\lambda= \left( \frac{\ln (4gL)}{Cgl^2} \right)^{1/2} .
\]
Then the bound becomes
\[
M(\alpha) \le 4C^{1/2} g^{1/2}l \left( \ln (4gL) \right)^{1/2},
\]
and this holds for $\alpha$'s from a set of $(2\pi)^{-g}d\alpha$
measure at least $1/2$.
The proof of Theorem \ref{T3.2} is complete. $\square$

Moreover, by re-examining the reasoning of this section, we
see that we can also prove the more general result already
mentioned. We can obtain
a whole series of pointwise approximations to $V_{\alpha}(x)$.
More specifically, the difference
between $f(\phi_x\alpha)$ and those terms of
$\sum b(m)\sin(m\cdot \phi_x\alpha+\varphi_m)$ for
which $|m|_1\le M$ is
\[
\lesssim g^{1/2}l (ld^{-1}\ln g)^M \left( \ln (gL) \right)^{1/2}
+ gl (ld^{-1}\ln g)^{N+1}
\]
for suitable $\alpha$. In other words, the Fourier series
of $f$ (viewed as a function on the Jacobi variety), up to some
order, gives a very good pointwise approximation to $V_{\alpha}(x)$
with positive probability (in fact, with as large probability as
we please) if $\alpha$ is chosen at random.

With $M=0$, Theorem \ref{T3.2} is recovered.
We do not need
these more refined statements to prove Theorem \ref{T1.1}.

\section{Proof of Theorem \ref{T1.1}}
The basic idea of the construction of \cite{Remex} was to
glue together suitably chosen periodic potentials. In this
paper, we will instead use finite gap
potentials with gaps of equal length.
Roughly speaking, the construction runs as follows.
We will choose the first finite gap potential $V_1$ so that
all gaps lie in, let us say, $[1,2]$. $V_2$
will have much smaller gaps; also, these new gaps will be
contained in the gaps of $V_1$.
If we continue in this way, the intersection
over all $n$ of the unions of the gaps of $V_n$ will be
a Cantor type set whose dimension is easily controlled,
provided there is an appropriate scaling. Moreover, the
set $S$ defined in \eqref{defS} will contain this Cantor type
set because if the energy $E$ is in a gap of $V_n$, the
solutions to the Schr\"odinger equation are on average
exponentially increasing or decreasing and hence do not
satisfy \eqref{WKB}. Of course, we must also take care of
the required decay of $V(x)$, that is, $V_n$ must be sufficiently
small for large $n$. The bounds on $V_n$ will be established with
the aid of Theorem \ref{T3.2}.

We start by investigating the solutions of \eqref{se} for
finite gap potentials $V$ and energies $E$ which lie in some
gap of $V$.
\begin{Lemma}
\label{L6.1}
Let $V(x)$ be a finite gap potential whose parameters satisfy
the assumptions of Theorem \ref{T3.2}. Then there exists an
$\epsilon=\epsilon(C_1,C_2,N)>0$ and a constant
$C=C(C_1,C_2,N)$, such that for
$ld^{-1}\ln g<\epsilon$, the following holds. If $|E-m_n| \le l_n/2$
for some $n\in \{ 1,\ldots, g\}$, then there is a solution
$y(x)$ of the Schr\"odinger equation \eqref{se} with
$y(x_0)=1$ for some $x_0\in [0,1]$ and
\[
\int_{x_0}^{\infty} |y(x)|^2 \, dx \le C/l_n .
\]
\end{Lemma}
{\it Remark.} This statement cannot, in general, hold with a
fixed, prescribed $x_0$ because the decaying solution has zeros.
Roughly speaking, the lemma says that there is a solution which
has some decay over intervals of length $\gg l_n^{-1}$.

{\it Proof.} Our starting point is the following formula
(see, for example,
\cite[Chapter 9]{CL}):
\begin{equation}
\label{m}
\int_x^{\infty} |f(t,z)|^2 \, dt = \frac{\text{Im }m_x(z)}{\text{Im }z}.
\end{equation}
Here, $m_x$ is the $m$-function of $-d^2/dt^2 + V(t)$ on
$[x,\infty)$ with Dirichlet boundary conditions at $t=x$. More
specifically, let $u,v$ be the solutions of $-y''+Vy=zy$ with the
initial values $u(x,z)=v'(x,z)=1$, $u'(x,z)=v(x,z)=0$ and write
\[
f(t,z) = u(t,z)+m_x(z) v(t,z);
\]
then $m_x(z)$ is defined by requiring that $f\in L_2(x,\infty)$.

The Green function of the whole line problem
is related to the $m$-function by
$G(x,x;z)=(m_x^{-}(z)-m_x(z))^{-1}$, where $m_x^{-}$ is the
$m$-function of the operator on $L_2(-\infty,x)$ (see again
\cite{CL}). Since the imaginary parts of
$m_x^{-}$ and $m_x$ have opposite signs,
the right-hand side of \eqref{m} is less than
$-\text{Im }G(x,x;z)^{-1}/\text{Im }z$. So, if we use \eqref{Green}
and abbreviate $\prod (\mu_j(x)-z) = U_x(z)$, then \eqref{m} becomes
\begin{equation}
\label{m1}
\int_x^{\infty} |f(t,z)|^2 \, dt < -\frac{2}{\text{Im }z} \,
\text{Im }\frac{R(z)}{U_x(z)} .
\end{equation}
Here, the sign of $R(z)$ is determined by
the fact that $\text{Im }G(x,x;z)$ has the same sign
as $\text{Im }z$ for
$\text{Im }z\not= 0$ (compare the discussion following \eqref{Green}).

Now let $E$ be as in the hypothesis, and put $z=E+i\delta$ with
$\delta>0$. By slightly changing $x$ if necessary, we may assume
that $\mu_n(x)\not= E$. Then $E$ is not in the spectrum of the
operator on $L_2(x,\infty)$ with Dirichlet boundary conditions.
This is so simply because $\mu_n(x)$ is the only eigenvalue
in the gap $(E_{2n-1},E_{2n})$. Thus $m_x(z)$ and $R(z)/U_x(z)$
are holomorphic in a neighborhood of $z=E$. For this latter function,
this may of course be seen by direct inspection. Moreover,
$R(E)/U_x(E)$ is real. Therefore, the right-hand side of
\eqref{m1} converges to $-2(R/U_x)'(E)$ as $\delta\to 0+$, while
the function $f(t,E+i\delta)$ tends to $f(t,E)=u(t,E)+m_x(E)v(t,E)$.
Fatou's Lemma together with
\eqref{m1} imply
\[
\int_x^{\infty} |f(t,E)|^2\, dt \le -2 \, \frac{d}{dz} \left(
\frac{R}{U_x} \right) (E) .
\]
We have that
$f(x,E)=1$,
so it remains to evaluate $(R/U_x)'(E)$. To this end, note that
\[
\left( \ln \frac{R}{U_x} \right)'(E) = \frac{1}{2(E-E_0)}
+\sum_{j=1}^g \left( \frac{1}{\mu_j(x)-E} - \frac{1}{2(E_{2j-1}-E)}
- \frac{1}{2(E_{2j}-E)} \right).
\]
Estimating as in the proof of Lemma \ref{L3.1}, we see that
\[
\sum_{j\not= n} \left|
\frac{1}{\mu_j(x)-E} - \frac{1}{2(E_{2j-1}-E)}
- \frac{1}{2(E_{2j}-E)} \right| \lesssim ld^{-2} .
\]
Furthermore, the term with $j=n$ can be bounded by $C/|\mu_n(x)-E|$.
Since this is $\gtrsim l_n^{-1}$ which is much larger than
$ld^{-2}$, we can in fact estimate the whole logarithmic derivative
by $C/|\mu_n(x)-E|$. Finally, similar arguments show that
$|(R/U_x)(E)| \lesssim l_n/|\mu_n(x)-E|$, so we conclude that
\[
\int_x^{\infty} |f(t,E)|^2\, dt \le \frac{Cl_n}{(\mu_n(x)-E)^2} .
\]
The proof is finished by observing that $\mu_n(x)$ moves by an
amount $\gtrsim l_n$ if $x$ varies over an interval of length
one. Indeed, if we again use the variables $\psi_j$ (see
\eqref{subst1}, \eqref{subst2}), then the $\psi_j$'s evolve
according to the differential equations
\[
\frac{d\psi_n}{dx} = \frac{2i R_n(m_n-l_n\cos\psi_n)}
{\prod_{j\not= n} (m_j-m_n-l_j\cos\psi_j+l_n\cos\psi_n)},
\]
and the right-hand sides are $\approx 1$, independently of the
positions the $\psi_j(x)$'s. $\square$

Now let $a_1=0$, $a_{n+1}=a_n+L_n$, where $L_n>0$ will be chosen
later. Then $V$ will be of the form
\[
V(x) = \sum_{n=1}^{\infty}\chi_{(a_n,a_{n+1})}(x) V_n(x-a_n);
\]
the building blocks $V_n$ are finite gap potentials.

We now pick these $V_n$'s. We basically keep the notation
of the preceding sections, except that there is now an additional
index $n$. The gaps of $V_n$ are taken to be of equal length
$l_n$, and
$g_n$ denotes the number
of gaps of $V_n$. Let $\alpha\in (1/2,1)$ be the exponent from
\eqref{hyp} (if $\alpha=1$, there is nothing to prove). We abbreviate
$2(1-\alpha)=D$, so $D\in (0,1)$, and $D$ is the dimension the
set $S$ from \eqref{defS} must have. Fix a number $a>(1-D)^{-1}$
and put
\[
l_n = \exp(-a^n).
\]
A Cantor type set with $g_n$ intervals of length $l_n$ as its
$n$th approximation has dimension $D$ if there is a scaling of
the type $g_nl_n^D\sim 1$. This suggests to take $g_n\sim
\exp(Da^n)$, but for technical reasons, the actual definition
is slightly different. First of all, choose a sequence $\epsilon_n>0$
which tends to zero, but so slowly that $\epsilon_na^n-
\epsilon_{n-1}a^{n-1}\to \infty$ and $a^n\exp(-\epsilon_n a^n)
\to 0$.
(In fact, we could take
$\epsilon_n=q^n$ with $a^{-1}<q<1$ right away, but if we leave
$\epsilon_n$ unspecified, we can improve the decay
rate of $V$.) Now put $g_{n_0-1}=1$, where $n_0\in\mathbb N$ must
be sufficiently large (this will be made precise below), and then
define inductively for $n\ge n_0$
\[
g_n= \exp\left( (D-\epsilon_n)a^n \right) + \theta_n g_{n-1}.
\]
Here, we require that $g_n/g_{n-1}\in \mathbb N$, and this
determines $\theta_n\in [0,1)$ uniquely.

The parameter $d_n$ describes the spacing
between adjacent gaps of $V_n$. Since we want these gaps to
lie in some gap of $V_{n-1}$,
we get $g_n \lesssim g_{n-1}l_{n-1}/d_n$.
If we disregard the $\epsilon_n$'s and $\theta_n$'s
for the moment and take $d_n$
about as large as possible,
this motivates the choice
\[
d_n = \exp\left( (-D +(D-1)a^{-1}) a^n \right) .
\]
A computation then shows that $l_nd_n^{-1} \ln g_n\to 0$,
so Theorem \ref{T3.2} can indeed be used to bound $V_n$ for
sufficiently large $n$.
Finally, Lemma \ref{L6.1} suggests that we put $L_n=A/l_n$,
with a large constant $A$ (how large $A$ has to be will become
clear later on).

It remains to choose the
locations of the gaps, that is, we must pick,
for each $n$, the parameters $E_0^{(n)}<m_n(1)<\cdots < m_n(g_n)$.
Let $G_{n_0-1}=[1,2]$, say
(any compact subinterval of $(0,\infty)$ will do).
For the time being, set $E_0^{(n_0)}=0$.

Pick $g_{n_0}$ different centers $m_{n_0}(1)<\cdots< m_{n_0}(g_{n_0})$
that lie well inside $[1,2]$ and satisfy
\[
\min_k (m_{n_0}(k)-m_{n_0}(k-1))\ge d_{n_0}.
\]
By the first requirement we mean that
$|m_{n_0}(k) - 3/2| \le 1/4$ for all $k=1,\ldots,
g_{n_0}$. Since $d_{n_0} g_{n_0}\to 0$ as $n_0\to\infty$,
this can certainly be done,
provided $n_0$ is large enough.

Let $G_{n_0}$ be the union of (the middle halves of) the corresponding
gaps:
\begin{equation}
\label{gn}
G_{n_0} = \bigcup_{k=1}^{g_{n_0}} \left[ m_{n_0}(k) - l_{n_0}/2,
m_{n_0}(k) + l_{n_0}/2 \right] .
\end{equation}

The general step is similar. Suppose $G_{n-1}$ has been constructed
for some $n\ge n_0+1$.
More specifically, $G_{n-1}$ is then a union of
(half the) gaps of $V_{n-1}$. For every such interval
$[m_{n-1}(j)-l_{n-1}/2,m_{n-1}(j)+l_{n-1}/2]\subset
G_{n-1}$, pick $g_n/g_{n-1}$ centers $m_n(k)$ with $|m_n(k)-m_{n-1}(j)|
\le l_{n-1}/4$, and also so that
the $m_n(k)$ are separated from one another by a distance $\ge d_n$,
Again, $m_n(k)$'s
with these properties exist in sufficiently large supply: Indeed,
we can find
$\approx l_{n-1}/d_n$ centers $m_n(k)$ with the required properties
in each subinterval of $G_{n-1}$. On the other hand,
by our assumptions on the sequence $\epsilon_n$,
the desired number of centers $g_n/g_{n-1}=(l_{n-1}/d_n)
e^{\epsilon_{n-1}a^{n-1}-\epsilon_n a^n} +\theta_n$ is much smaller
than $l_{n-1}/d_n$ for all $n\ge n_0+1$, provided $n_0$ is large enough.
So the construction is possible. We then
get a total of $(g_n/g_{n-1})g_{n-1}=g_n$ new gaps,
and we would like to define $G_n$
as the union of these gaps, as in \eqref{gn}.

Unfortunately, there is an additional complication: If we
proceed as above, we have no control on $\widehat{f}_0$,
so the bound of Theorem \ref{T3.2} is useless. The problem
is that we have not renormalized correctly. So it is necessary to
also adjust $E_0^{(n)}$. We will add $E_0^{(n)}$ to
all the previously chosen centers $m_n(k)$ ($k=1,\ldots,g_n$),
while keeping the gap length $l_n$ fixed. This just
amounts to adding $E_0^{(n)}$ to the function $f(\alpha)$.
In particular, the original mean value $\widehat{f}_0$ will
be replaced by $\widehat{f}_0+ E_0^{(n)}$.

Now \eqref{f0} implies that (for $E_0^{(n)}=0$)
$\widehat{f}_0= O(g_nl_n)$, so for a suitable $E_0^{(n)}= O(g_nl_n)$,
we have $\widehat{f}_0 = 0$, as desired.
On the other hand, since we were cautious enough
to take the gaps of $V_n$ well inside the gaps of $V_{n-1}$,
the new centers $E_0^{(n)}+m_n(k)$ will still satisfy, let us say,
$|E_0^{(n)} + m_n(k) - m_{n-1}(j)|
\le l_{n-1}/3$. This follows from the fact
that the shift $E_0^{(n)}=O(g_nl_n)$ is much smaller than $l_{n-1}$
for large $n$ (here
we use the inequality $a>(1-D)^{-1}$).

Once $E_0^{(n)}$ has been picked,
$G_n$ can again be defined as in \eqref{gn}, but with the shifted
centers $E_0^{(n)}+m_n(k)$ taking the role of $m_n(k)$.

This two step procedure (first choose $m_n(k)$'s, then shift
by an appropriate $E_0^{(n)}$ to make $\widehat{f}_0 = 0$) can
now be used to pick the $m_n(k)$'s and $E_0^{(n)}$ (inductively)
for all $n\ge n_0$. Note
that the construction ensures that $G_n\subset G_{n-1}$.

We must still choose, for every $n\ge n_0$,
a particular potential from the corresponding family $V_{\alpha_0}$
of finite gap potentials. Fortunately, this choice is easy:
We fix once and for all a sufficiently large $N\in\mathbb N$
(where ``sufficiently large'' will be made precise at the end
of the proof) and then simply take a $V_n$ that satisfies the
conclusion of Theorem \ref{T3.2} for $L=L_n$.
Note also that the assumptions
of Theorem \ref{T3.2}
on the location of the gaps (that is, $C_1 \le m_n(k)-E_0^{(n)}\le C_2$
for all $k=1,\ldots,g_n$)
hold with $n$-independent constants $C_1,C_2>0$
because $E_0^{(n)}$ lies
in a small interval centered at zero while all gaps are in
$[1,2]$. Therefore, the constant $C$ from the statement of
Theorem \ref{T3.2} is also independent of $n$.

Finally, for $n<n_0$, we can take an arbitrary bounded (and
measurable) function as $V_n$; for instance, we can put
$V_n\equiv 0$ for $n<n_0$.

Let $T=\bigcap_n G_n$. We will now show that $S\supset T$,
where $S$ is the set defined in \eqref{defS}.
Suppose $E\in T$. Then $E$ satisfies the assumptions of Lemma \ref{L6.1}
for the potentials $V(x)=V_n(x)$ for all
sufficiently large $n$. Assume, to obtain
a contradiction, that $E\notin S$. Then there is a solution
$y(x)$ of the form \eqref{WKB}, and $y,\overline{y}$ span the
space of solutions of \eqref{se}. In particular, the solution
$f_n$ from Lemma \ref{L6.1} must be a linear combination of
$y$ and $\overline{y}$, that is,
\begin{equation}
\label{lsg}
f_n(x) = A_n e^{i\omega(x)} + B_n e^{-i\omega(x)} + r_n(x),
\end{equation}
with $\omega(x)=\int_0^x \sqrt{E-V(t)}\, dt$ and
$|r_n(x)| \le (|A_n|+|B_n|) \rho (x)$, where $\rho(x)\to 0$. Of
course, since $V=V_n$ only on the interval $(a_n,a_{n+1})$,
eq.\ \eqref{lsg} holds for $a_n\le x\le a_{n+1}$.
Lemma \ref{L6.1} shows that $f_n(x_0^{(n)})=1$ for some
$x_0^{(n)}\in [a_n,a_n+1]$ and
\begin{equation}
\label{contra}
\int_{x_0^{(n)}}^{a_{n+1}} \left| f_n(x)\right|^2 \, dx
\le C/l_n
\end{equation}
for large values of $n$.
We also take $n$ so large that
$\rho(x) < \delta$ for
$x\ge a_n$ and $\|V_n\|_{\infty}<\delta E$,
which clearly is possible since $g_nl_n \to 0$ (of course, Theorem
\ref{T3.2} would give a better bound on $V_n$, but this is not needed
here). Then, since $f_n(x_0^{(n)})=1$, we must have that
$|A_n|+|B_n| > (1+\delta)^{-1}$. Now routine estimates show
that if $\delta>0$ was chosen sufficiently small,
then \eqref{lsg} implies that
\[
\int_{x_0^{(n)}}^{a_{n+1}} \left|f_n(x)\right|^2 \, dx \ge C_0 L_n
= AC_0/l_n.
\]
This inequality contradicts \eqref{contra} if $A$ is sufficienctly
large,
so $T\subset S$, as claimed.

The next step is to prove that $\dim T = D$.
To this end, we introduce a Borel measure $\mu$ that reflects
the self-similar scaling structure of $T$. More specifically,
$\mu$ gives equal weight to the intervals of
$G_n$ for every $n$: $\mu(I_n)= g_n^{-1}$ if $I_n$ is one
of the intervals $[m_n(k)-l_n/2, m_n(k)+l_n/2]$.
Moreover, we also demand that $\mu$ be
supported by $T$: $\mu(\mathbb R \setminus T)=0$. It is not
hard to show (for instance, by considering approximations
$\mu_n$ supported by $G_n$) that there indeed exists a unique
Borel (probability) measure $\mu$ satisfying these requirements.


We will now establish
the following property of the generalized derivatives of $\mu$:
For every fixed $\gamma < D$, we have that
\begin{equation}
\label{deri}
\lim_{\delta\to 0+} \sup_{|I|\le \delta} \frac{\mu(I)}{|I|^{\gamma}}
=0 .
\end{equation}
The supremum is over all intervals $I\subset\mathbb R$ of length
at most $\delta$.

If \eqref{deri} holds, then, by general facts on Hausdorff measures
\cite[Section 3.4, Theorem 67]{Rog}, $\mu$ gives zero weight
to sets of dimension strictly less than $D$, and therefore
$\dim T\ge D$, as desired. The converse inequality $\dim T\le D$
does not need explicit proof (although that would actually be
easy to do) because we know that always $\dim S\le 2(1-\alpha)$
(this is the result whose optimality we are about to prove), and
thus $\dim T\le\dim S\le D$ will follow automatically once we have
established that $V(x)=O(x^{-\alpha})$.

So let us prove \eqref{deri}:
Fix $\gamma$, and let $I$ be an interval with $|I|\le\delta$,
where $\delta>0$ is small. Then, define $n\in\mathbb N$ by
requiring that $l_n<|I|\le l_{n-1}$. Clearly, $n$ is large if
$\delta$ is small. We first treat the
case when $|I| \le d_n$. Recall that $d_n$ is
the minimal distance between adjacent
gaps of $V_n$. So the above assumption implies that $I$ intersects
at most two of the intervals that build up $G_n$. Each
of these intervals has measure $g_n^{-1}$, hence
\[
\frac{\mu(I)}{|I|^{\gamma}} \le \frac{2}{g_n |I|^{\gamma}}
\le \frac{2}{g_n l_n^{\gamma}}.
\]

On the other hand, if $|I|> d_n$, then the number of
subintervals of $G_n$ intersecting $I$ is $\le 3|I|/d_n$,
thus in this case,
\[
\frac{\mu(I)}{|I|^{\gamma}} \le \frac{3|I|^{1-\gamma}}{d_ng_n}
\le 3\, \frac{g_{n-1}l_{n-1}}{d_ng_n}\,
\frac{1}{g_{n-1}l_{n-1}^{\gamma}} .
\]
Now $g_nl_n^{\gamma} \gtrsim \exp( \sigma a^n)$, where
$\sigma>0$ depends on $\gamma$, and $g_{n-1}l_{n-1}/(d_n
g_n) \lesssim \exp(\epsilon_n a^n)$; indeed, relations of this
type motivated our definition of $g_n$ and $d_n$. Since, as noted
above, $n\to\infty$ as $\delta\to 0+$, \eqref{deri} now follows.

It remains to show that $V$ satisfies the bound \eqref{hyp}.
So, let $x\in (a_n,a_{n+1})$ with large $n$. Recall that
$\widehat{f}_0=0$, where $f$ is the function from the trace
formula for $V_n$. Theorem \ref{T3.2} therefore implies that
\[
\left|V(x) \right| \le C \left[ g_n^{1/2} l_n \left(\ln (g_nL_n)
\right)^{1/2} +
g_nl_n (l_nd_n^{-1}\ln g_n)^{N+1} \right]
\]
for these $x$. On the other hand,
\[
x  \le a_{n+1} =\sum_{m=1}^n L_m
\lesssim \sum_{m=1}^n l_m^{-1} \lesssim l_n^{-1},
\]
and \eqref{hyp} indeed follows, provided we took
\[
N+1\ge \frac{1-\alpha}{2\alpha - 1}\, \frac{a}{a-1}.
\]
(As expected, $N\to\infty$ as $\alpha\to 1/2+$.) $\square$

Actually, doing these final estimates carefully,
we obtain a stronger bound of the form
$x^{-\alpha-\epsilon_n/2}(\ln x)^{1/2}$.
The strengthening of Theorem
\ref{T1.1} mentioned in Sect.\ 1 follows from this by taking
a sequence $\epsilon_n$ that tends to zero sufficiently
slowly.

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