% Zero Measure Spectrum for the Almost Mathieu Operator
% This is a (plain) TeX document
\baselineskip 20pt
\magnification 1200
\mathsurround=1pt
\font\bigrm=cmr10 scaled\magstep3
\def \spec {{\rm Spec\,}}
\def \spr {{\rm Spr\,}}
\def \intb {{\rm Intb\,}}
\def \imag {{\rm Im\,}}
\def \real {{\rm Re\,}}
\def \xvec {\vec {\bf x}}
\def \yvec {\vec {\bf y}}
\def \uvec {\vec {\bf u}}
\def \irrh {H_{\alpha, \lambda, \theta}}
\def \rath {H_{{p/q}, \lambda, \theta}}
\def \discr {D_{{p/q}, \lambda, \theta}(E)}
\def \udis {\Delta_{{p/q}, \lambda}(E)}
\def \irrsp {\sigma (\alpha, \lambda, \theta)}
\def \ratsp {\sigma ({p/q}, \lambda, \theta)}
\def \crsp {\sigma (\alpha, 2, \theta)}
\def \irrusp {S(\alpha, \lambda)}
\def \irrg {G(\alpha, \lambda)}
\def \irrisp {S_- (\alpha, \lambda)}
\def \ratusp {S({p_n/q_n}, \lambda)}
\def \ratgn {G({p_n/q_n}, \lambda)}
\def \ratisp {S_- ({p/q}, \lambda)}
\def \crusp {S({p/q}, 2)}
\def \dimh {{\rm dim}_{\rm H}}
\def \cntl {\centerline}
\def\no{\noindent}
\def\gap{\bigskip\no}
\def\ref#1#2#3#4#5#6{#1:\ #2.\ #3\ {\bf #4},\ #5 \ (#6) }
\def \aaa {1}
\def \ids {2}
\def \avs {3}
\def \blt {4}
\def \bas {5}
\def \cham {6}
\def \cey {7}
\def \cad {8}
\def \cyc {9}
\def \del {10}
\def \harp {11}
\def \hesj {12}
\def \hofst {13}
\def \falc {14}
\def \fsw {15}
\def \nmt {16}
\def \ysp {17}
\def \wilk {18}
\def \srv {19}
\def \sinai {20}
\def \tha {21}
\def \thb {22}
\def \tata {23}
\def \tatb {24}
\def \tod {25}
\def \wats {26}
\hfill March 10, 1993
\bigskip
\bigskip
\cntl{\bigrm Zero Measure Spectrum}
\cntl{\bigrm for the Almost Mathieu Operator
\footnote*{\rm Work partially supported by the GIF.}}
\bigskip \vskip 2.0cm \cntl{ Y. Last}
\cntl{Department of Physics} \cntl{Technion -
Israel Institute of Technology}
\cntl{32000 Haifa, Israel}
\cntl{E-mail: phr97yl@technion.bitnet}
\bigskip
\vskip 2.3cm
\noindent
{\bf Abstract. } We study the almost Mathieu operator:
$(H_{\alpha,\lambda,\theta}u)(n)=
u(n+1)+u(n-1)+\lambda\cos (2\pi\alpha n+\theta)u(n)$, on $l^2(Z)$, and show
that for all $\lambda,\theta$, and (Lebesgue) a.e.\ $\alpha$, the Lebesgue
measure of its spectrum is precisely $|4-2|\lambda||$. In particular, for
$|\lambda|=2$ the spectrum is a zero measure cantor set. Moreover, for a large
set of irrational $\alpha$'s (and $|\lambda|=2$) we show that the Hausdorff
dimension of the spectrum is smaller than or equal to $1/2$.
\vfill\eject
\noindent
{\bf 1. Introduction }
\medskip \noindent
In this paper, we study the almost Mathieu (also called Harper's) operator on
$l^2(Z)$. This is the (bounded, self adjoint) operator $\irrh$, defined by:
$$\eqalign{\irrh=H_0+V_{\alpha,\lambda,\theta} \; ,&\qquad
(H_0u)(n)=u(n+1)+u(n-1) \; ,\cr
(V_{\alpha,\lambda,\theta} u)(n)&=\lambda\cos (2\pi\alpha n+\theta)u(n)
\; ,\cr} \eqno (1.1)$$
where $\alpha,\lambda,\theta \in R$.
$\irrh$ is a tight binding model for the Hamiltonian of an electron in a one
dimensional lattice, subject to a commensurate (if $\alpha$ is rational) or
incommensurate (if $\alpha$ is irrational) potential. It is also related to
the Hamiltonian of an electron in a two dimensional lattice, subject to a
perpendicular magnetic field [\harp,\hofst] (in which case the relevant energy
spectrum is the union over $\theta$ of the energy spectra of $\irrh$).
The almost Mathieu operator has been studied by many authors
[\aaa { }- \hofst,\fsw,\ysp { }- \tatb,\wats],
and many of its spectral characteristics are known. Our main result in this
paper is:
\gap
{\bf Theorem 1. } {\it If $\alpha$ is an irrational, for which there is a
sequence of rationals $\{p_n/q_n\}$ obeying:
$$\lim_{n\to\infty}\;{q_n}^2\Bigl|\alpha-{p_n\over {q_n}}\Bigr|\;=\;
0 \; ,$$
then for every $\lambda, \theta \in R$:
$$|\irrsp| = |4-2|\lambda|| \; ,$$
where $\irrsp$ is the spectrum of $\irrh$, and $|\cdot|$ denotes Lebesgue
measure.
\gap
Remarks.} 1) The set of irrationals characterized in Theorem 1 is precisely
the set of irrationals having unbounded continued fraction expansions. This
set is known to have full Lebesgue measure [\nmt].
\no
2) The $\theta$ independence part of Theorem 1 is immediate, since, for
irrational $\alpha$, $\irrsp$ itself is known to be independent of $\theta$
[\cyc].
\bigskip
The equality $|\irrsp|=|4-2|\lambda||$ was conjectured by Aubry and Andre
[\aaa] to hold for every irrational $\alpha$. It was later studied by
Thouless [\tha], and by Avron,
van Mouche, and Simon [\avs], who established the inequality
$|\irrsp|\geq |4-2|\lambda||$ (for every $\alpha,\lambda,\theta$).
For $|\lambda|\not= 2$, Theorem 1 has already been proved in [\ysp].
The main theme
of the current paper is the handling of the case $|\lambda|=2$, for which we
prove:
\gap
{\bf Lemma 1. } {\it Let $p,q\in N$ be relatively prime, and denote:
$$\irrusp\equiv\bigcup_\theta\irrsp\; ;$$ then:
$${{2(\sqrt5+1)}\over q}\; < \;|S({p/q}, 2)|\; < \;{{8e}\over q}$$
(where $e\equiv\exp(1) =2.71.\,.\,.$).}
\gap
It should be remarked that a similar (though somewhat weaker) lower bound on
$|\crusp|$ was already established in [\wilk]. It is the upper bound in Lemma
1, which is the main new result of the current paper and from which the
completion of the proof of theorem 1 follows.
Lemma 1 is strongly related to a conjecture of Thouless
[\tha,\thb,\tata,\tatb], which says:
$$\lim_{q\to\infty}\, q|\crusp|\; = \; {\rm Const} \; = \; 9.32.\,.\,.$$
This conjecture was found numerically for sequences with $p,q$
relatively prime and $q\to\infty$. It was derived analytically only for some
sequences with fixed $p$ and $q\to\infty$ [\thb,\tata,\wats]. Some
(nonrigorous) analytical argumentation that it should also hold for more
general cases was given in [\wilk].
It is interesting to note that since $\irrsp$ has no isolated points [\cyc],
the vanishing of its measure for $|\lambda|=2$ also implies:
\gap
{\bf Corollary 1.1. } {\it For irrational $\alpha$ as in
Theorem 1, $\crsp$ is a (zero measure) Cantor set (i.e. a closed, nowhere
dense set, with no isolated points).}
\gap
Moreover, if $\alpha$ is an irrational which is very well approximated by
rationals, we will show that Lemma 1 implies an upper bound on the Hausdorff
dimension of $\crsp$, namely:
\gap
{\bf Theorem 2. } {\it If $\alpha$ is an irrational obeying:
$${q_n}^4\Bigl|\alpha-{p_n\over {q_n}}\Bigr|\;<\; C\; ,$$
for some constant $C$, and a sequence of rationals $\{p_n/q_n\}$ with
$q_n\to\infty$, then:
$$\dimh(\crsp)\;\leq\; {1\over 2} \; ,$$
where $\dimh(\cdot)$ denotes Hausdorff dimension.
\gap
Remark. } The set of irrationals characterized in Theorem 2 has zero Lebesgue
measure, but it contains a dense $G_\delta$ set, which makes it ``generic'' in
the commonly used topological sense.
\bigskip
The analysis leading to our results is based on previous findings of Avron,
van Mouche, and Simon [\avs], and this paper is, to a large extent, a
continuation of their work. While some parts of this analysis were already
carried out in [\ysp] and in [\wilk], for the reader's convenience, we shall
repeat the relevant derivations of those papers.
In Section 2 we describe some preliminaries and previously obtained results.
In Section 3 we prove Lemma 1, and in Section 4 we prove Theorem 1. Finally,
in Section 5, we prove Theorem 2.
It is a pleasure to thank J. Avron, B. Simon, and M. Wilkinson for useful
discussions.
\medskip\gap\gap
{\bf 2. Preliminaries}
\medskip\no
We begin this section with a remark about the considered ranges of $\alpha,
\lambda,\theta$. Since $\irrh$ is invariant under: $\alpha\to\alpha\pm 1$,
$\theta\to\theta\pm 2\pi$, we may always assume: $\alpha\in [0,1]$,
$\theta\in [0,2\pi]$. This has no effect on the correctness of our results
for more general values of $\alpha$ and $\theta$. Moreover, a sign change
of $\lambda$ ($\lambda\to -\lambda$) is equivalent to a translation of
$\theta$ by $\pi$. Thus, any quantity or result which is independent of
$\theta$ must be invariant under a sign change of $\lambda$, and throughout
the rest of the paper we will usually assume: $\lambda\geq 0$.
In what follows we will be largely concerned with the spectral analysis of
the almost Mathieu operator at rational frequencies. That is, we will consider
$\irrh$ where $\alpha=p/q$, $p,q\in N$, and we assume throughout that $p$
and $q$ are relatively prime (i.e. they have no common divisor other than 1).
In this case $\irrsp$ does depend on $\theta$, and we will also be interested
in the two spectral sets:
$$\eqalign{\irrusp &\;\equiv\;\bigcup_\theta \irrsp\cr\noalign{\medskip}
\irrisp &\;\equiv\;\bigcap_\theta \irrsp\; . \cr} \eqno (2.1)$$
These sets are also well defined for irrational $\alpha$, but in this case:
$\irrisp=\irrusp=\irrsp$. As we shall see later, the set $\irrusp$ has good
continuity properties (in $\alpha$), and our results for irrational $\alpha$
are essentially based on the study of $\irrusp$ for rational $\alpha$.
A central role in the spectral analysis of $\rath$ is played by the
discriminant $\discr$, defined by:
$$\discr\;\equiv\;{\rm Trace}\left[\pmatrix{E-V(1)&-1\cr\noalign{\medskip}1&0}
\pmatrix{E-V(2)&-1\cr\noalign{\medskip}1&0}\;\cdots\;
\pmatrix{E-V(q)&-1\cr\noalign{\medskip}1&0}\right]\; ,
\eqno (2.2)$$
where $V(n)\equiv\lambda\cos(2\pi(p/q)n+\theta)$. $\discr$ is a polynomial of
order $q$ (in $E$) having the following properties (see e.g. [\tod]):
\medskip
\item{(i)} $\discr$ has $q$ real simple zeroes.
\item{(ii)} $\discr$ is larger than or equal to 2 at all its maxima points,
and it is smaller than or equal to -2 at all its minima points.
\medskip\no
The spectrum $\ratsp$ is precisely the inverse image under $\discr$ of the
interval $[-2,2]$ (i.e. it is precisely the set of $E$'s for which:
$-2\leq\discr\leq 2$). Thus, from the properties of $\discr$ it is seen that
$\ratsp$ is made of $q$ bands (closed intervals), such that
$\discr$ is
strongly monotone on each band. A remarkable formula, originally due to
Chambers [\cham] (also see [\bas] for a proof), gives the $\theta$ dependence
of $\discr$:
\gap
{\bf Proposition 2.1. } {\it If $p,q$ are relatively prime, then:
$$\discr\;=\;\udis - 2{{\left({\lambda\over 2}\right)}^q}\cos\theta q$$
where $\udis\equiv {D_{{p/q}, \lambda, \pi/2q}(E)}$.}
\gap
Proposition 2.1 implies that $S({p/q},\lambda)$ is precisely the
inverse image under
$\udis$ of the interval $[-2-2(\lambda/2)^q\,,\;2+2(\lambda/2)^q]$. Moreover,
it shows that if $\lambda > 2$ then $\ratisp=\emptyset$, and if
$\lambda\leq 2$ then $\ratisp$ is the inverse image under $\udis$ of the
interval $[-2+2(\lambda/2)^q\,,\;2-2(\lambda/2)^q]$. We remark that from the
fact that the above properties (i) and (ii) of $\discr$ hold for every
$\theta$, and from Proposition 2.1, it follows that $\udis$ is larger than or
equal to $2+2(\lambda/2)^q$ at all its maxima points, and it is smaller than
or equal to
$-2-2(\lambda/2)^q$ at all its minima points. Moreover, each of the sets
$S({p/q},\lambda)$ and $\ratisp$ (when it is not empty) is made of $q$
bands, such
that $\udis$ is strongly monotone on each band.
An important property of $\irrh$ is the Aubry duality [\aaa], which allows
relating eigenfunctions and spectra of $\irrh$ to those of
$H_{\alpha, {4/\lambda}, \theta}$. The following version of this duality was
rigorously proven by Avron and Simon [\ids]:
\gap
{\bf Proposition 2.2. } {\it For every real $\alpha$:
$$\irrusp\;=\;{\lambda\over 2}S(\alpha, {4/\lambda})\; .$$}
\gap
Thus, it is sufficient to study $\irrusp$ for $0\leq\lambda\leq 2$, since,
for $\lambda > 2$, $\irrusp$ is obtained by Proposition 2.2 from the
$\lambda <2$ case.
Avron, van Mouche, and Simon [\avs] proved the following:
\gap
{\bf Proposition 2.3. } {\it For $0\leq\lambda\leq 2$ and $p,q$ relatively
prime:
$${\rm (i)}\qquad\qquad\qquad\qquad\qquad\quad |\ratisp|\;=\;4-2\lambda\; .
\qquad\qquad\qquad\qquad\quad$$
$${\rm (ii)}\;\;\,\qquad\qquad\qquad 4-2\lambda\;\leq\;
|S({p/q},\lambda)|\;\leq\;4-2\lambda +4\pi{\left(\lambda\over
2\right)}^{q/ 2}\; .
\qquad\quad\;\;\,$$}
\gap
In particular, Proposition 2.3 shows that if $0\leq\lambda < 2$, and if
$p_n/q_n \to\alpha$, where the $p_n/q_n$'s are rationals, and $\alpha$ is
irrational, then $|S({p_n/q_n},\lambda)|\to 4-2\lambda$. If $\lambda=2$ then
statement (ii) of Proposition 2.3 becomes useless; but, it was shown in
[\wilk] that, in this case, the remarkable exact equality for $|\ratisp|$
(statement (i)) translates to an exact equality involving the slopes of
$\udis$ at its zero crossings. Namely:
\gap
{\bf Proposition 2.4. } {\it If $p,q$ are relatively prime, then:
$$\sum_{\nu=1}^q{1\over|\Delta'_{{p/q}, 2}(E_\nu)|}\;=\;{1\over q}\; ,$$
where $\Delta'_{{p/q}, \lambda}(E)\equiv
{d\over dE}\Delta_{{p/q}, \lambda}(E)$, and $E_1,E_2,\dots ,E_q$ are the
$q$ zeroes of $\Delta_{{p/q}, 2}(E)$.
\gap
Proof. } Since $\ratisp$ is the inverse image under $\udis$ of the interval
$[-2+2(\lambda/2)^q\,,\;2-2(\lambda/2)^q]$, and since $\udis$ is also a
polynomial in $\lambda$, we have, for $\lambda < 2$, in the limit
$\lambda\to 2$:
$$|\ratisp|\;\;\sim\;\;
\sum_{\nu=1}^q{{4-4{({\lambda/ 2})}^q }\over
|\Delta'_{{p/q}, \lambda}(E_\nu)|}\; . \eqno (2.3)$$
Thus, from statement (i) of Proposition 2.3 we obtain:
$$\sum_{\nu=1}^q{1\over|\Delta'_{{p/q}, 2}(E_\nu)|}\;=\;
\lim_{\lambda\nearrow 2}\,{{|\ratisp|}\over
{4-4{\left({\lambda/ 2}\right)}^q}}\;=\;\lim_{\lambda\to 2}\,
{{4-2\lambda}\over{4-4{\left({\lambda/ 2}\right)}^q}}\;=\;{1\over q}\; .
\;\;\bigcirc\eqno (2.4)$$
\gap
Proposition 2.4 is in the heart of Lemma 1 that we prove in the next Section.
For every $\alpha,\lambda\in R$, the set $\irrusp$ is compact, and, therefore,
it has definite edges: $\max\irrusp\,,\,\min\irrusp\in\irrusp$. The
complement of $\irrusp$ in the interval $[\min\irrusp\,,\,\max\irrusp]$ is
open, and it is therefore a union of countably many (finite) open intervals.
We shall refer to such open intervals, when they are chosen to have maximal
length, as gaps in $\irrusp$, and we shall denote their union by $\irrg$.
That is:
$$\irrg\;\equiv\;[\min\irrusp\,,\,\max\irrusp]\;\setminus\;\irrusp\; ,
\eqno (2.5)$$
and so we have:
$$|\irrusp|\;=\;\max\irrusp - \min\irrusp - |\irrg|\; . \eqno (2.6)$$
When $\alpha$ is rational ($\alpha = p/q$) we have seen that $\irrusp$ is
made of $q$ bands. Thus, it has at most $q-1$ gaps. When $\alpha$ is
irrational $\irrusp$ may have an infinite number of gaps.
We conclude this Section by quoting another result of Avron, van Mouche, and
Simon [\avs], this time regarding the continuity properties of $\irrusp$:
\gap
{\bf Proposition 2.5. } {\it
For every $\lambda>0$, there is a constant $C$, such that
if $|\alpha-\alpha'| E_2^\nu$ , and $E_0^\nu$
is inside the gap just above $I_\nu$).
Define:
$$f(E)\;\equiv\;{d\over {dE}}(\ln (\Delta(E))) \; . \eqno (3.1)$$
Since $\Delta(E)$ can be expressed as:
$$\Delta(E)\; = \;\prod_{j=1}^q (E-E_j) \; , \eqno (3.2)$$
$f(E)$ can be written as:
$$ f(E)\; = \;\sum_{j=1}^q {1\over{E-E_j}} \; , \eqno (3.3)$$
and we have:
$$f'(E) \;\equiv\; {d\over {dE}} f(E)\; = \;
- \sum_{j=1}^q {1\over {(E-E_j)^2}} \; . \eqno (3.4)$$
>From (3.4) we see that:
$$f'(E)\; < \;{{-1}\over {(E-E_\nu)^2}} \; , \eqno (3.5)$$
and since $E_0^\nu$ is a zero of $f(E)$ , we have for every $E \in (E_\nu ,
E_0^\nu )$:
$$f(E)\;=\; - \int_E^{E_0^\nu} f'(E')\, dE'
\;>\; \int_E^{E_0^\nu} {{dE'}\over {(E'-E_\nu)^2}}
\;=\; {1\over {E-E_\nu}} - {1\over {E_0^\nu - E_\nu}} \; .
\eqno (3.6)$$
Now, consider $E\in (E_\nu, E_2^\nu)$ . Since $\Delta(E_2^\nu) = 4$ , we
have:
$$\ln {4\over {\Delta(E)}}\; = \;\ln \Delta(E_2^\nu) - \ln \Delta(E)\; =
\;\int_E^{E_2^\nu} f(E')\, dE' \; , \eqno (3.7)$$
and by using (3.7) and the estimate (3.6), we obtain:
$$\ln {4\over {\Delta(E)}}\; > \;\ln\left({{E_2^\nu-E_\nu}\over {E-E_\nu}}
\right) - 1 \; . \eqno (3.8)$$
(3.8) implies:
$${4\over {\Delta(E)}}\; > \;
{1\over e}\left({{E_2^\nu-E_\nu}\over {E-E_\nu}}\right)\; , \eqno (3.9)$$
which can also be written as:
$$E_2^\nu - E_\nu\; < \;4e{{E-E^\nu}\over {\Delta(E)}} \; . \eqno (3.10)$$
Since $\Delta(E_\nu) = 0$, we obtain from (3.10), by letting $E\to E_\nu$:
$$E_2^\nu - E_\nu\; < \;{{4e}\over {\Delta'(E_\nu)}} \; . \eqno (3.11)$$
Clearly, a similar calculation can also be carried for the lower part of
the band, by integrating $f'(E)$ from the minimum of $\Delta(E)$ just below
(or at) $E_1^\nu$ . Thus, we also have:
$$E_\nu - E_1^\nu\; < \;{{4e}\over {\Delta'(E_\nu)}} \; , \eqno (3.12)$$
which (together with (3.11)) implies:
$$|I_\nu|\; = \; E_2^\nu - E_1^\nu\; < \;{{8e}\over {\Delta'(E_\nu)}} \; .
\eqno (3.13)$$
It's easy to see that (3.13) could be obtained with a similar calculation,
also if $\Delta(E)$ was decreasing on $I_\nu$. Thus, (3.13) clearly holds
for all of the nonextermal bands. If $I_\nu$ is an
extermal band, we can still make a similar calculation to (3.1)-(3.11)
for the ``less extermal'' part of this band, and obtain either (3.11)
(for the lowest band) or (3.12) (for the highest band). But, since
$|\Delta'(E)|$ is monotone on an extermal band, the ``more extermal''
part of such a band must be smaller then its ``less extermal'' part.
Thus, (3.13) holds for every band, and from Proposition 2.4 we
obtain:
$$|S|\; =\; \sum_{\nu=1}^q|I_\nu|\; < \;{{8e}\over q} \; .
\;\;\bigcirc \eqno (3.14)$$
\gap
(ii) {\it Proof of the lower bound. } For each band of $S$:
$I_\nu = [E_1^\nu , E_2^\nu]$, we denote the two parts of the band by:
$$b_1^\nu\;\equiv\;[E_1^\nu,E_\nu]\;,\qquad
b_2^\nu\;\equiv\;[E_\nu,E_2^\nu]\; . \eqno (3.15)$$
Since $\Delta'(E)$ is a polynomial
of order $q-1$, which has $q-1$ distinct real zeroes, $|\Delta'(E)|$ has
a single maximum between every two consecutive zeroes of $\Delta'(E)$, and
it is monotone above and below the extreme zeroes of $\Delta'(E)$.
Thus, $|\Delta'(E)|$ has a single maximum on each band (which may be at
the edge of the band), and it is monotone on every subinterval of the
band which does not contain this maximum. This implies that for each
band, either for all $E\in b_1^\nu$ (if the maximum is on $b_2^\nu$),
or for all $E\in b_2^\nu$ (if the maximum is on $b_1^\nu$), we have:
$$|\Delta'(E)|\; \leq \;|\Delta'(E_\nu)|\; . \eqno (3.16)$$
Since:
$$\int_{E_1^\nu}^{E_\nu}|\Delta'(E)|\,dE\; = \;
|\Delta(E_\nu)-\Delta(E_1^\nu)|\; = \; 4 \eqno (3.17)$$
and also
$$\int_{E_\nu}^{E_2^\nu}|\Delta'(E)|\,dE \;= \;
|\Delta(E_2^\nu)-\Delta(E_\nu)|\; = \; 4 \; , \eqno (3.18)$$
we have, either for $i=1$, or for $i=2$:
$$|b_i^\nu|\; > \;{4\over {|\Delta'(E_\nu)|}}\;\equiv\; l_\nu\; .
\eqno (3.19)$$
In [\wilk], (3.19) has been used to obtain: $|S|>4/q$. We shall now improve
this bound.
Suppose that $I_\nu$ is a nonextermal band and that $\Delta(E)$ is
increasing on $I_\nu$, and Consider the polynomial:
$$G(E) \;\equiv\; \Delta(E) + 4 \; , \eqno (3.20)$$
which has a zero at $E_1^\nu$ . A similar estimate to (3.1)-(3.6)
shows that for every $E\in (E_1^\nu , E_0^\nu)$:
$${{G'(E)}\over {G(E)}}\; = \;{d\over {dE}}\ln G(E)
\; > \;{1\over {E-E_1^\nu}} - {1\over {E_0^\nu - E_1^\nu}} \; .
\eqno (3.21)$$
By taking $E=E_\nu$ in (3.21), we obtain:
$${{\Delta'(E_\nu)}\over 4}\; = \;{{G'(E_\nu)}\over {G(E_\nu)}}\; > \;
{1\over {|b_1^\nu|}} - {1\over {|b_1^\nu| + |b_2^\nu|}}\; = \;
{{|b_2^\nu|}\over {|b_1^\nu| (|b_1^\nu| + |b_2^\nu|)}} \; , \eqno (3.22)$$
which implies:
$$|b_1^\nu|\; > \;{{l_\nu |b_2^\nu|}\over {|b_1^\nu| + |b_2^\nu|}}\; .
\eqno (3.23)$$
Clearly, by considering: $H(E) \equiv \Delta(E) - 4$ instead of $G(E)$,
and integrating from the minimum of $H(E)$ just below $E_1^\nu$ ,
we can similarly obtain (3.23) with $b_1^\nu$ and $b_2^\nu$ interchanged,
namely:
$$|b_2^\nu|\; > {\;{l_\nu |b_1^\nu|}\over {|b_1^\nu|+|b_2^\nu|}}\; .
\eqno (3.24)$$
We have seen that either $|b_1^\nu| > l_\nu$ or
$|b_2^\nu| > l_\nu$ . Suppose that $|b_2^\nu| > l_\nu$, then (3.23)
implies:
$$|b_1^\nu|\; > \;{{l_\nu}^2\over {|b_1^\nu|+l_\nu}} \; , \eqno (3.25)$$
which can be rewritten as:
$$|b_1^\nu|^2 + |b_1^\nu| l_\nu - {l_\nu}^2\; > \;0 \; . \eqno (3.26)$$
Solving the appropriate quadratic equation shows that (3.26) implies:
$$|b_1^\nu|\; > \;{{\sqrt5 - 1}\over 2} l_\nu \; . \eqno (3.27)$$
Similarly, if $|b_1^\nu| > l_\nu$ , (3.24) would imply:
$$|b_2^\nu|\; > \;{{\sqrt5 - 1}\over 2} l_\nu \; , \eqno (3.28)$$
so in either case we have:
$$|I_\nu|\; = \;|b_1^\nu| + |b_2^\nu|\; > \;{{\sqrt5 + 1}\over 2}l_\nu \; ,
\eqno (3.29)$$
and it is clear that (3.29) holds for every nonextermal band. For
the case of an extermal band, only one of the inequalities (3.23)
or (3.24) can be obtained. But, in this case, due to the monotonicity of
$|\Delta'(E)|$, we also know which one
of the $|b_i^\nu|$ $(i=1,2)$ is larger and obeys: $|b_i^\nu| > l_\nu$ .
It is easy to verify that $|b_2^\nu| > l_\nu$ corresponds to the case
where (3.23) holds, and that the other case corresponds to (3.24). Thus,
we obtain (3.29) for all the bands, and from Proposition 2.4 we get:
$$|S|\; > \;{{2(\sqrt5 + 1)}\over q} \; . \;\;\bigcirc \eqno (3.30)$$
\medskip\gap\gap
{\bf 4. Proof of Theorem 1.}
\gap
{\bf Lemma 4.1. } {\it For every $\lambda\in R$, and a sequence of
rationals $\{p_n/q_n\}$, with $p_n,q_n$ relatively prime, and $q_n\to\infty$:
$$\lim_{n\to\infty}|S({p_n/q_n}, \lambda)|\;=\;|4-2|\lambda||$$
\gap
Proof. } Combining statement (ii) of Proposition 2.3 and Lemma 1 we obtain
for every $0\leq\lambda\leq 2$: $|S({p_n/q_n},\lambda)|\to 4-2\lambda$.
>From that and from Proposition 2.2 the Lemma follows. $\bigcirc$
\gap
{\bf Proposition 4.1. } (Thouless [\tha]; Avron, van Mouche, and Simon
[\avs])
\no
{\it For any irrational $\alpha$, and $\lambda,\theta\in R$:
$$|\irrsp|\;\geq\; |4-2|\lambda||\; .$$
\gap
Proof. } Let $\{g_j(\alpha,\lambda)\}_{j=1}^\infty$ be the gaps in $\irrusp$
(ordered somehow), and pick some $\epsilon > 0$. Since $\sum_{j=1}^\infty
|g_j(\alpha,\lambda)| = |\irrg|$, there is a finite $J_\epsilon$ such that
$\sum_{j=1}^{J_\epsilon} |g_j(\alpha,\lambda)| > |\irrg|-\epsilon$. Now,
consider a sequence of rationals: $p_n/q_n\to\alpha$. From statement (i) of
Corollary 2.1 we have:
$$\liminf_{n\to\infty}|\ratgn|\;\geq\;\sum_{j=1}^{J_\epsilon}
|g_j(\alpha,\lambda)|\; > \;|\irrg|-\epsilon\; , \eqno (4.1)$$
and from statement (ii):
$$\lim_{n\to\infty}\,\matrix{\max\cr\min\cr}S({p_n/q_n},\lambda)\;\; = \;\;
\matrix{\max\cr\min\cr}S(\alpha,\lambda)\; . \eqno (4.2)$$
Thus, from (2.6) we obtain:
$$|\irrusp|\; > \;\limsup_{n\to\infty}|S({p_n/q_n},\lambda)| -\epsilon\; ,
\eqno (4.3)$$
which by Lemma 4.1 implies:
$$|\irrusp|\; > \; |4-2|\lambda||-\epsilon\; . \eqno (4.4)$$
Since $\irrusp = \irrsp$ and since $\epsilon$ is arbitrary this completes
the proof. $\bigcirc$
\gap
{\it Proof of Theorem 1. } Let $\alpha$ be an appropriate irrational, and let
$\{p_n/q_n\}$ be a sequence of rationals obeying:
$\lim_{n\to\infty}\,{q_n}^2|\alpha-{p_n/q_n}| = 0$. Obviously, $q_n\to\infty$,
and we can assume $p_n,q_n$ to be relatively prime.
Since there
are at most $q_n-1$ gaps in $S({p_n/q_n},\lambda)$, we obtain from statement
(i) of Corollary 2.1 (for $|\alpha-{p_n/q_n}|\;|G({p_n/q_n},\lambda)|-
12(q_n-1)(\lambda|\alpha-{p_n/q_n}|)^{1/2}\; . \eqno (4.5)$$
By (2.6) and statement (ii) of Corollary 2.1 this implies:
$$|S(\alpha,\lambda)|\;<\;|S({p_n/q_n},\lambda)|+
12q_n(\lambda|\alpha-{p_n/q_n}|)^{1/2}\; . \eqno (4.6)$$
As $n\to\infty$, we have from Lemma 4.1:
$|S({p_n/q_n},\lambda)|\to |4-2|\lambda||$, and by our assumption on
$\{{p_n/q_n}\}$: $q_n|\alpha-{p_n/q_n}|^{1/2}\to 0$. Thus, (4.6) implies:
$$|\sigma(\alpha,\lambda,\theta)|\;=\;|S(\alpha,\lambda)|\;\leq\;
|4-2|\lambda||\; , \eqno (4.7)$$
which together with Proposition 4.1 completes the proof.
$\bigcirc$
\medskip\gap\gap
{\bf 5. Proof of Theorem 2. }
\gap
{\bf Lemma 5.1. } {\it Let $S\subset R$, and suppose that $S$ has a sequence
of covers: $\{S_n\}_{n=1}^\infty$, $S\subset S_n$, such that each $S_n$ is
a union of $q_n$ intervals, $q_n\to\infty$ as $n\to\infty$, and for each $n$:
$$|S_n|\; < \; {C\over {{q_n}^\beta}}\; , $$ where $\beta$ and $C$
are positive constants; then:
$$\dimh (S)\; \leq\; {1\over {1+\beta}} \; .$$
\gap
Proof. } Let $S_n = \bigcup_{\nu=1}^{q_n}{b_\nu^n}$, where each $b_\nu^n$
is an interval, and let $t=1/(1+\beta)$, then:
$${1\over{q_n}}\sum_{\nu=1}^{q_n}{|b_\nu^n|}^t \;\leq\;
\left({1\over{q_n}}\sum_{\nu=1}^{q_n}{|b_\nu^n|}\right)^t \; ,
\eqno (5.1)$$
which implies:
$$\sum_{\nu=1}^{q_n}{|b_\nu^n|}^t\;\leq\;
{q_n}^{1-t}\left(\sum_{\nu=1}^{q_n}{|b_\nu^n|}\right)^t\; = \;
{q_n}^{1-t}|S_n|^t\;\leq\;
{q_n}^{1-t}\left({C\over {{q_n}^\beta}}\right)^t\; = \;
{q_n}^{1-(1+\beta)t} C^t\; = \;
C^t \; . \eqno (5.2)$$
Recall that the Hausdorff dimension of $S$ is given by (see e.g. [\falc]):
$$\dimh(S)\; = \; \inf \left\{\,t\in{R^+}\;\Bigg|\;\lim_{\delta\to 0}\;
\inf_{\delta{\hbox{\sevenrm -covers}}}\sum_\nu|b_\nu|^t\; < \;\infty\right\} \;
\eqno (5.3)$$
where a ${\delta{\hbox{\rm -cover}}}$ is a cover of $S$:
$S\subset\bigcup_{\nu=1}^\infty b_\nu$, such that each $b_\nu$ is an interval, a
$|b_\nu|<\delta$. Thus, since $q_n\to\infty$ as $n\to\infty$, (5.2) implies:
$\dimh(S)\leq 1/(1+\beta)$. $\bigcirc$
\gap
{\it Proof of Theorem 2. } Let $\alpha$ be an appropriate irrational, and let
$\{p_n/q_n\}$ be a sequence of rationals obeying: $q_n\to\infty$ as
$n\to\infty$, and ${q_n}^4|\alpha-p_n/q_n|