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\centerline{\bf Summability of large powers of logarithm of classic lacunary series}
\centerline{\bf and its simplest consequences.}
\bigskip
This paper can be considered as a small appendix to my large article [1] in
``Algebra \& Analysis''. I'm going to discuss here the last result on
lacunary series absolute value distribution obtained there, namely, the
following
\bigskip
\noindent \underbar{\bf Theorem 1.}
\medskip
Let $\varepsilon > 0$ be arbitrarily small.
\medskip
Let $\Lambda =\{ m_k\}_{k=-\infty}^{\infty} \subset \bb Z$ satisfy the Zygmund
lacunarity condition, i.e.
$$\sup_{r \neq 0} \ {\rm card}\ \{(k', k'') : m_{k'} - m_{k''} = r\} \
{\buildrel {\rm def}\over =} R(\Lambda) < + \infty. \qquad\qquad(*)$$
Then for any function $f \epsilon L^2 ({\bb T})$ spectred at $\Lambda$ and for
each subset $E \subset {\bb T}$ of positive measure the inequality
$$\| f \|_{L^2 ({\bb T})}^2 \leq \exp \Big\{ {B(\varepsilon,
R(\Lambda))\over \mu (E)^{2 + \varepsilon }} \Big\} \int_E \big| f\big|^2 d
\mu$$
holds with some constant $B(\varepsilon, R(\Lambda))$ depending on $\varepsilon$
and $R(\Lambda)$ only. As it has already been mentioned in [1], the power $2 +
\varepsilon$ in the denominator in the exponent cannot be replaced by
$1-\varepsilon$ without additional assumptions on $\Lambda$ even if we restrict
ourselves to arcs $E$ instead of arbitrary measurable sets. Nevertheless, it
was commonly believed that much stronger inequality must hold for the most
classical example of a spectrum $\Lambda$, satisfying $(*)$: that with Hadamard
gaps. In particular, it was conjectured that any (arbitrarily large) power of
$log| f |$ is integrable while the theorem 1 implies
the summability of powers less than ${1\over 2}$ only. The following
modification of theorem 1 allows to prove this conjecture.
\bigskip
\noindent \underbar{\bf Theorem 2.}
\medskip
Under the conditions of theorem 1, suppose in addition that the difference set
$D\Lambda {\buildrel {\rm def}\over =}$\hfill\break $m_{k'}-m_{k''}: k' \neq
k''\}$ is a
$\Lambda (p) $ - system with $p > 2$ (recall that a set $U \subset {\bb Z}$ is
called a $\Lambda (p)$ - system if every function $g \ \epsilon\ L^2 ({\bb T})$
spectred at $U$ belongs to $L^p ({\bb T})$ as well and $\| g
\|_{L^p ({\bb T})} \leq C \| g\|_{L^2 (\bb T)}$ with some
constant $c >0$ not depending on
$g$).
\bigskip
Then for any function $f \ \epsilon\ L^2 ({\bb T})$ spectred at $\Lambda$
and for
each subset $E \subset {\bb T}$ of positive measure the inequality
$$\| f\|_{L^2 ({\bb T})}^2 \leq \exp \Big\{{B\over \mu (E)^{{4\over p} +
\varepsilon}}\Big\} \int\limits_E |f|^2 d\mu$$
holds with some constant $B=B(\varepsilon, \Lambda) > 0$ not depending on $f$
and $E$.
\bigskip
\noindent \underbar{\bf Proof}:
\medskip
The following proposition was the crucial point in the proof of the theorem 1.
\bigskip
\noindent \underbar{\bf Main lemma}:
\medskip
Let $\Lambda \subset {\bb Z}$ satisfy the Sigmund lacunarity condition, $E
\subset {\bb T}$, $\mu (E) \leq {1\over 2}, \ \ f \epsilon\ L^2 ({\bb T})$ and
$spec \,f \subset \Lambda$.
\medskip
If a positive integer $n$ is so large that
$${4\over \mu (E')^2} \sum_{\buildrel i,j \epsilon {\bb Z}\over {i\neq j}} |
\hat{\chi}_{{}_{E'}} (m_i - m_j)|^2 \leq n + 1$$
for each subset $E' \subset E$ of measure $\mu (E') \geq {\mu(E)\over 2}$,
then there exists a set $E_1 \supset E$ such that
\medskip
\item{1)} $\dis{\mu(E_1 \backslash E) \geq {\mu(E)\over 4n}}$
\medskip
\item{2)} $\dis{\int\limits_{E_1} |f|^2 d\mu \leq \Big\{{An^5\over \mu
(E)^2}\Big\}^{3n}
\int\limits_E | f |^2 d \mu}$ with some absolute constant $ A > 0$.
\bigskip
This lemma was one of the most complicated results in [1] and to outline its
full proof, one should rewrite about a half of 60 pages of [1]. So I have to
refer the reader to [1] for details. Nonetheless, once having been proved, it
is very easy to apply: we are merely to iterate its result until the measure of
the next set $E_k$, say, will get greater than ${1\over 2}$ and after that to
use the statement of theorem 1. The main problem is to make the parameter $n$
as small as possible at each step, i.e.~to get a good estimate of the sum
$\sum\limits_{i\neq j} |\hat{\chi}_{{}_{E'}} (m_i - m_j)|^2$. In [1] the trivial
estimate
$$\eqalign{& \sum_{i\neq j} |\hat{\chi}_{{}_{E'}} (m_i - m_j)|^2 =\sum_{r\neq 0}
|\hat{\chi}_{{}_{E'}} (r)|^2 card\,\{(i,y):m_i - m_j = r\} \leq \cr
& R(\Lambda) \sum_{r \epsilon {\bb Z}} |\hat{\chi}_{{}_{E'}} (r) |^2 = R
(\Lambda) \mu
(E')\cr}$$
was used, which allowed to take $n \sim {1\over \mu (E)}$. If we take into
account that only $r \epsilon D \Lambda$ may occur in the last sum and that for
any $\Lambda (p)-$system $U \subset {\bb Z}$
$$\eqalign{&\big\{\sum_{r \epsilon U} \big| \hat{\chi}_{{}_{E'}} (r) \big|^2
\big\}^{1\over 2} =\sup \Big\{ \Big|\int\limits_{{}_{E'}} \overline{g} d \mu
\Big| \ : \
\| g
\|_{L^2 ({\bb T})} \leq 1,\ \ {\rm spec}\ \ g \subset U\Big\} \leq \cr
&\leq \mu (E')^{1-{1\over p}} \sup \{\| g\|_{L^p ({\bb T})} :\
\| g\|_{L^2 ({\bb T})} \leq 1,\ \ {\rm spec}\ \ g \subset U\} \leq
C\mu (E')^{1-{1\over p}},\cr}$$
we shall get much better inequality
$$\sum\limits_{i\neq j} | \hat{\chi}_{{}_{E'}} (m_i - m_j)|^2 \leq R(\Lambda) C^2 \mu
(E')^{2-{2\over p}},$$
which makes possible to put $n =\big[{B\over \mu (E)^{2/p}}\big]$
with some $B > 0$ depending on $\Lambda$ only ($[x]$ stands here for the entire
part of a real number $x$, as usual).
\bigskip
Thus, if we denote by $C(\nu)$ the best possible constant, for which the
inequality\hfill\break $\| f \|_{L^2(T)}^2 \leq C(\nu)
\int\limits_E |f|^2 d\mu$ holds for every function $f \in L^2 ({\bb T})$
spectred at
$\Lambda$ and for every set $E \subset {\bb T}$ of measure $\mu (E) \geq
\nu$, and put
$\psi(\nu) \buildrel{\rm def}\over = \log C(\nu),\ \ \Delta
(\nu) \buildrel{\rm def}\over = {1\over 4B} \nu^{1 +{2\over p}}$, we
shall get
$$
{\psi(\nu) - \psi (\nu + \Delta(\nu))\over \Delta
(\nu)} \leq {12 B^2\over \nu^{1 + {4\over p}}} \log {AB^5\over
\nu^{2 + {10\over p}}} \leq {B_{\varepsilon} \over \nu^{1 +
{4\over p}
+ \varepsilon}}
$$
for each $0 < \nu \leq {1\over 2}$. Comparing this inequality with the
corresponding differential that ${d\psi\over d\nu} (\nu) \leq
{B_{\varepsilon}\over \nu^{1 + {4\over p} + \varepsilon}}$, we can easily
conclude that $\psi (\nu) = O (\nu^{-{4\over p} -
\varepsilon})$ as it
has been promised. It is well-known that a spectrum $\Lambda$ with Hadamard
gaps does actually satisfy the conditions of the theorem 2 with arbitrarily
large $p > 0$ and, therefore, we get the estimate
$$\| f\|_{L^2 ({\bb T})}^2 \leq
\exp\Big\{{B_{\varepsilon}\over \mu
(E)^{\varepsilon}}\Big\} \int\limits_E |f|^2 d\mu$$
with arbitrarily small $\varepsilon > 0$ and some $B_{\varepsilon} > 0$
depending on $\varepsilon$ and $\Lambda$ only (not on $f$ or $E$).
\bigskip
Moreover, we can make $p$ dependent on $\mu (E)$ when choosing $n$ and obtain
the better inequality.
$$\| f\|_{L^2({\bb T})}^2 \leq \exp \Big\{B \log^{10} {2\over \mu
(E)}\Big\} \int_E |f|^2 d\mu.$$
The power 10 isn't, of course, the best possible one, but it cannot be
replaced by any number less than 2 even for arcs $E$.
\bigskip
Another well-known exmaple, underlying the conditions of the theorem 2, is the
following. Let $p$ be an even integer great than 2, $\Lambda =\{m_k\}_{k
\in {\bb Z}}$ be such that there is no nontrivial linear dependence between
integers $m_k$ with integer coefficients for which the sum of the absolute
values of the coefficients is $2p$ or less (actually this is an example of a
spectrum for which the corresponding constant, connecting $L^p$ and $L^2$-norms
of a function spectred at the difference set, is the least possible one among
those for spectra with infinitely many elements). Such a spectrum can be
rather dense, namely it can contain about $N^{{1\over 2p} - \varepsilon}$
elements among first $N$ successive integers (to construct the corresponding
example, it is enough to consider a random spectrum $\Lambda =\{n \in {\bb
N} : \xi_n = 1\}$, where $\xi_n (n=1,2,\cdots)$ are independent random
variables defined by $P\{\xi_n = 1\} = \delta n^{{1\over 2p} - \varepsilon -
1},\ \ P\{\xi_n =0\} = 1 -\delta n^{{1\over 2p} - \varepsilon -1}$ with
$\delta >
0$ sufficiently small). According to the Mandelbrojt-Belov-Miheev theorem, this
means that the power $-{4\over p} - \varepsilon$ in the conclusion of the
theorem 2 cannot be replaced by $-{1\over 2p -1} + \varepsilon$ even for arcs
$E$ (at least for $p=4,6,\cdots$). The result of theorem 2 may be used in
different ways. We shall mention below only a couple of the most immediate
consequences; in both of them we suppose $\Lambda$ to satisfy the conditions of
the theorem 2 with some $p > 4$.
\bigskip
\noindent \underbar{\bf Corollary 3.}
\medskip
Let $\bar{a}=\{ a_k\}_{k=-\infty}^{\infty}$ be a sequence of complex numbers
for which $\sum\limits_{k=-\infty}^{\infty} |a_k|^2 < +\infty$ and $a_k =0$ for
every
$k \notin \Lambda$. Let $S$ be the left shift operator in $\ell^2 ({\bb Z})$
(i.e. $({\cal S} a)_i = a_{i+1},\ i \in {\bb Z}$). Then $a$ is not a
cyclic vector for ${\cal S}$.
\bigskip
\noindent \underbar{\bf Proof}:
\medskip
This is merely a reformulation of the obvious statement that either $a=0$
or
$\dis{\int\limits_{\bb T} \Big|\log \Big|\sum_{k \in {\bb Z}} a_k z^k \Big|\Big|
d\mu (z) < + \infty.}$
\bigskip
\noindent \underbar{\bf Corollary 4}.
\medskip
Let $\Lambda \subset {\bb Z}_{+}$ and $f(z) = \sum\limits_{k\in
\Lambda} a_k
z^k$ be an analytic function in the unit disk ${\bb D}$.
\bigskip
\item{(a)} if $\sum\limits_{k \in \Lambda} |a_k |^2 < + \infty$ then $f
\in H^2 ({\bb D})$ and has no singular factor in its canonical
factorization,
\bigskip
\item{(b)} If $\sum\limits_{k\in \Lambda} |a_k|^2 = + \infty$ then $f$ has
infinitely many zeros in ${\bb D}$, these zeros being everywhere dense near the
boundary of ${\bb D}$ and ``equidistributed'' there in the following sense: let
$$\tau (r)^2 \ {\buildrel {\rm def}\over = } \sum\limits_{k\in \Lambda}
|a_k|^2\ r^{2k} \ \ ( 0 < r < 1),\ \ \nu_r\ {\buildrel {\rm def}\over =}
\sum\limits_{\buildrel{\xi: f(\xi)=0}\over {0 < | \xi | < r}} \log {r\over
|\xi |}
\delta_{\xi}$$
(as usual $\delta_{\xi}$ denotes the unit mass concentrated at
$\xi$ and zeros are counted with their multiplicity); then ${1\over \log \tau
(r)} \nu_r$ weakly converges to the Lebesgue measure $\mu$ on the unit
circumference when $r \rightarrow 1-$.
\vfill\eject
\noindent \underbar{\bf Remark.}
\medskip
The statement $(b)$ is worth comparing with the well-known result due to
Murai claiming that for spectra $\Lambda \subset {\bb Z}_{+}$ with Hadamard
gaps the density of zeros near the boundary may be derived from much weaker
assumption $\sup\limits_{z\in {\bb D}} | f(z) | = + \infty$ (or, what
is the
same for series with Hadamard gaps,
$\sum\limits_{k \in \Lambda} |a_k | = + \infty)$. But Murai's proof uses
the Hadamard lacunarity condition in its full strength and I don't know whether
there is any essentially denser spectrum $\Lambda$ for which his result remains
valid.
\bigskip
\noindent \underbar{\bf Proof}:
\medskip
The part (a) follows immediately from the simple observation that either $f
\equiv 0$ or the integrals $\int\limits_{\bb T} | \log | f ( r z)
\|^{1+\delta} \ d \mu (z)$ are uniformly bounded for ${1\over 2} < r < 1$
if $\delta > 0$ is sufficiently small.
\bigskip
To prove the statement (b) it is enough to check that the measures $\eta_r
{\buildrel{\rm def}\over =} {1\over \log \tau (r)} \nu_r$ satisfy the
following two conditions:
\bigskip
\item{1)} for every $R < 1 \ \ \eta_r ({\bb D}_R) \rightarrow 0$
when $r\rightarrow 1-$ (here ${\bb D}_R {\buildrel {\rm def}\over =} \{ z: | z |
\leq R\}$.
\bigskip
\item{2)} if $|\xi | < 1$ and $f(\xi)\neq 0$, then $\int\limits_{\bf C}
{1-|\xi|^2\over | z - \xi |^2}\ d \eta_r (z)
\rightarrow 1$ when $r\rightarrow 1-$
\bigskip
\noindent (the set of linear combinations of Poisson kernels is dense in $C({\bb
T})$).
\bigskip
(1) is obvious because there are only finitely many zeros of $f$ within
${\bb D}_R$ and $\tau(r) \rightarrow + \infty$ when $r \rightarrow 1-$. When
checking (2) we may assume without loss of generality that $f(0) \neq 0$ (the
division of $f$ by $z^m$ changes nothing). If $r > |\xi |$ we can write the
Jensen formula
$$\sum_{{{\zeta: | \zeta | < r}\atop {f(\zeta )=0}}}\log
{ |r^2 -\overline{\xi} \zeta |\over r| \xi - \eta |} = \int_{{\bb T}} \log | f
(rz)| {r^2 - |\xi |^2 \over |\xi - rz|^2} d\mu (z) - \log |f(\xi)|.$$
The result of the theorem 2 implies immediately that the right hand part is
$\log \tau (r) + O(1)$ when $\xi$ is fixed and $r \rightarrow 1-$.
\medskip
\noindent Therefore
$$\int\limits_{{\bf D}} \log {| r^2 - \overline{\xi} z | \over r | z -\xi|} \ d
\eta_r (z)
\rightarrow 1 \quad {\rm when}\quad r \rightarrow 1-.$$
Note now that
$$\log {|r^2 - \overline{\xi} z|\over r | z-\xi|} = {1\over 2} \log \left( 1+
{r^2 - |\xi|^2\over | \xi - z|^2} \ \big(1-\big|{z\over r}\big|^2\big)\right).$$
The ratio ${r^2 - | \xi |^2\over | \xi - z|^2}$ is uniformly bounded for all $z
\in supp \,\eta_r$ (i.e.~for $z$ satisfying $f (z) =0)$; and the
ratio $\big|{z\over r}\big|$ is very close to 1 when $r$ is close to 1 and $| z
|$ is close to 1. Thus we can replace $log {|r^2 - \overline{\xi} z |\over r
| z - \xi|}$ by the equivalent expression ${1-|\xi |^2\over | z - \xi |^2}
log \big|{r\over z}\big|$, getting the result desired. Very similar
considerations allow also to prove the following
\bigskip
\noindent \underbar{\bf Corollary 5.}
\medskip
Fix any (small) $\delta > 0$ denote $T_{\delta} {\buildrel {\rm def}\over =}
\{x + iy: 1-\delta < x < 1,\ \ y < \delta ( 1-x)\}$. If $f$ satisfies the
conditions of the corollary 4 then $f(\zeta T_{\delta})$ is dense in ${\bb C}$
for almost all $\zeta \ \in\ {\bb T}$.
\medskip
\noindent {\bf Literature cited:}
\medskip
\item{[1]}
\medskip
\item{[2]}
\medskip
\item{[3]}
\end