# On the complete separation of unique $\ell_{1}$ spreading models and the
Lebesgue property of Banach spaces

Harrison Gaebler  ,  Pavlos Motakis  and  Bünyamin Sarı  Department of
Mathematics, University of North Texas, 1155 Union Circle 311430 Denton, Texas
76203-5017  Department of Mathematics and Statistics, York University, 4700
Keele Street, Toronto, Ontario, M3J 1P3, Canada<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We construct a reflexive Banach space $X_{\mathcal{D}}$ with an unconditional
basis such that all spreading models admitted by normalized block sequences in
$X_{\mathcal{D}}$ are uniformly equivalent to the unit vector basis of
$\ell_{1}$, yet every infinite-dimensional closed subspace of
$X_{\mathcal{D}}$ fails the Lebesgue property. This is a new result in a
program initiated by Odell in 2002 concerning the strong separation of
asymptotic properties in Banach spaces.

###### 2020 Mathematics Subject Classification:

46B06, 46B20, 46B25, 46G12

The second author was supported by NSERC Grant RGPIN-2021-03639.

##  1\. Introduction

In Banach space theory, an asymptotic notion refers to a concept describing
the behavior of the norm of linear combinations of vectors sampled in a
specified asymptotic fashion from a certain type of structure. For example,
the seminal notion of a spreading model of a bounded sequence
$(x_{n})_{n=1}^{\infty}$ is defined as the unit vector basis
$(e_{i})_{i=1}^{\infty}$ of $c_{00}$ with the seminorm given by

$\Big{|}\sum_{i=1}^{n}a_{i}e_{i}\Big{|}=\lim_{k_{1}\to\infty}\lim_{k_{2}\to\infty}\cdots\lim_{k_{n}\to\infty}\Big{\|}\sum_{i=1}^{n}a_{i}x_{k_{i}}\Big{\|},$

provided that this iterated limit exists for all choices of coefficients
$a_{1},\ldots,a_{n}$. Here, the structure in question is a bounded sequence of
vectors in a Banach space, and vectors are sampled over a sparse set of
sufficiently large indices. Using Ramsey’s Theorem, Brunel and Sucheston
proved in [9] that every bounded sequence in a Banach space has a subsequence
generating some spreading model. Similar notions about different types of
structures have been introduced and studied over the years. One such example
is the notion of asymptotic models, introduced by Odell and Halbeisen in [13],
concerning the norm of linear combinations of vectors sampled from an array of
sequences. A different type of such a notion is asymptotic spaces, introduced
by Maurey, Milman, and Tomczak-Jaegerman in [17], concerning the norm of
linear combinations of vectors picked in a finite-round two-player game
between a vector chooser and a finite-codimensional subspace chooser. The
different types of asymptotic notions are relevant to the study of many
subjects in Banach space theory, such as the distortion problem in [18] using
asymptotic spaces, the invariant subspace problem in reflexive Banach spaces
in [4] using spreading models, the uniform approximation properties of bounded
linear operators in [2] using joint spreading models, the coarse geometry of
Banach spaces in [7] using asymptotic models, unique maximal ideals in the
operators of stopping time-type Banach spaces in [16] using spreading models,
and quotients algebras of operator spaces in [20] using asymptotic versions.
The wide applicability of these tools has led to an independent study of their
uniqueness properties and a desire to better understand their
interconnections. Among the most outspoken promoters of this program was
Odell, who in this context posed several questions of the following type (see
[22], [23], and [15]): Let $P$, $Q$ be properties of Banach spaces that are
stable under passing to closed subspaces and assume that $P$ implies $Q$. If a
Banach space $X$ satisfies Q must it have an infinite-dimensional closed
subspace satisfying $P$? For such $P$ and $Q$, if there exists a Banach space
$X$ satisfying $Q$ and all infinite-dimensional subspaces of which fail $P$
then we say that $P$ is completely separated from $Q$. For a detailed
discussion on positive and negative results in this context, we refer the
reader to [5]. We prove such a complete separation theorem for $P$ the
Lebesgue property and $Q$ the property of admitting a unique $\ell_{1}$
spreading model.

Let $X$ be a Banach space. A function $f:[0,1]\to X$ is Riemann-integrable if
it is bounded and continuous up to a set of Lebesgue measure zero (see, e.g.,
[12, Theorem 18] or [10, Theorem. 2.1.3.]). However, the converse statement is
false in general. Indeed, it is easy to check that the function which maps the
rationals in $[0,1]$ to the unit vector basis of $c_{0}$ is Riemann-
integrable. A Banach space $X$ is therefore said to have the Lebesgue property
if every Riemann-integrable function $f:[0,1]\to X$ is continuous up to a set
of Lebesgue measure zero. The first and third named authors recently in [11]
characterized the Lebesgue property in terms of a new asymptotic notion. In
particular, a Banach space $X$ has the Lebesgue property if and only if every
normalized basic sequence in $X$ is Haar-$\ell_{1}^{+}$ (see [11] or Section 2
of this paper for the precise definition of Haar-$\ell_{1}^{+}$). It is also
proved in [11] that for the properties:

  1. (P1)

every asymptotic model of $X$ is equivalent to the unit vector basis of
$\ell_{1}$

  2. (P2)

$X$ satisfies the Lebesgue property

  3. (P3)

every spreading model of $X$ is equivalent to the unit vector basis of
$\ell_{1}$

the implications (P1)$\implies$(P2)$\implies$(P3) are true. On the other hand,
(P2)$\centernot{\implies}$(P1) because the Tsirelson sum of Tsirelson spaces,
$(T\oplus T\oplus\ldots)_{T}$, has the Lebesgue property but no unique
asymptotic model (see, e.g., [11]) and (P3)$\centernot{\implies}$(P2) because
there exist Schur spaces that do not have the Lebesgue property (e.g.
[14],[21]) and Schur spaces satisfy (P3). The complete separation of (P1) from
(P2) was proved in [11] by showing that the Banach space $X_{iw}$ constructed
in [5] has the Lebesgue property, and yet, every subspace of $X_{iw}$ contains
a $c_{0}$-asymptotic model. The complete separation of (P2) from (P3) was not
attempted in [11], and it is the purpose of the present article. By carrying
out a substantial modification of the construction in [5], we are able to
achieve this new result. We construct a Banach space $X_{\mathcal{D}}$ with an
unconditional basis such that every spreading model of $X_{\mathcal{D}}$ is
uniformly equivalent to the unit vector basis of $\ell_{1}$ and such that
every subspace of $X_{\mathcal{D}}$ fails the Lebesgue property.

Let us explain the main properties of $X_{\mathcal{D}}$. Denote by
$\mathcal{D}$ the binary tree $\cup_{n=0}^{\infty}\\{0,1\\}^{n}$. Following
standard convention, $\\{0,1\\}^{0}$ is a singleton containing the empty
sequence. For $n\in\mathbb{N}\cup\\{0\\}$ and $\lambda\in\\{0,1\\}^{n}$ we
write $|\lambda|=n$ and say $\lambda$ has height $n$. We consider
$\mathcal{D}$ with the usual partial order with minimum the empty sequence
and, otherwise, $\lambda\leq\mu$ means $|\lambda|\leq|\mu|$ and
$\lambda(i)=\mu(i)$, for $1\leq i\leq|\lambda|$. The characterization of the
Lebesgue property in terms of Haar-$\ell_{1}^{+}$ sequences from [11] easily
yields the following (see Remark 2.4): a Banach space $X$ has the Lebesgue
property if and only if for every collection
$(x_{\lambda})_{\lambda\in\mathcal{D}}$ of norm-one vectors in $X$ there exist
$\theta>0$ and, for every $n\in\mathbb{N}$, a collection
$(\mu_{\lambda})_{\lambda\in\\{0,1\\}^{n}}$ such that, for
$\lambda\in\\{0,1\\}^{n}$, $\mu_{\lambda}\geq\lambda$ and

$\frac{1}{2^{n}}\Big{\|}\sum_{\lambda\in\\{0,1\\}^{n}}x_{\mu_{\lambda}}\Big{\|}\geq\theta.$

In the space $X_{\mathcal{D}}$, every infinite-dimensional subspace contains a
lexicographically block collection $(x_{\lambda})_{\lambda\in\\{0,1\\}^{n}}$
of norm-one vectors such that for any $n\in\mathbb{N}$ and collection
$(\mu_{\lambda})_{\lambda\in\\{0,1\\}^{n}}$ in $\mathcal{D}$ such that
$\mu_{\lambda}\geq\lambda$, $\lambda\in\\{0,1\\}^{n}$,

(1)
$\max_{\lambda\in\\{0,1\\}^{n}}|a_{\lambda}|\leq\Big{\|}\sum_{\lambda\in\mathcal{D}^{n}}a_{\lambda}x_{\mu_{\lambda}}\Big{\|}\leq
3\max_{\lambda\in\mathcal{D}^{n}}|a_{\lambda}|,$

for any scalars $(a_{\lambda})_{\lambda\in\mathcal{D}^{n}}$. Therefore,
$X_{\mathcal{D}}$ fails the Lebesgue property. At the same time, every
normalized weakly null sequence in $X_{\mathcal{D}}$ has a subsequence
generating a spreading model that is $102$-equivalent to the unit vector basis
of $\ell_{1}$. In particular, it from follows James’s characterization of
reflexivity for spaces with an unconditional basis that $X_{\mathcal{D}}$ is
reflexive.

Our construction relies on the method of saturation under constraints with
increasing weights, also employed in [5]. The major difference lies in the
involvement of an additional metric constraint. A distantly related version of
this constraint was also present in [19], but its application and role were
significantly different as it produced an operator-algebraic outcome (see [19,
Section 2]). The earliest form of saturation under constraints is due to Odell
and Schlumprecht (see [24] and [25]). It has since been extensively developed,
e.g., in the papers [4], [8], [5], and many others. Let us elaborate on the
main ingredients of our construction. The norm of the space $X_{\mathcal{D}}$
is defined via a norming set, i.e., there is a subset $W$ of $c_{00}$
containing the unit vector basis $(e_{i}^{*})_{i=1}^{\infty}$ such that, for
$x\in c_{00}$, $\|x\|=\sup\\{|\langle f,x\rangle|:f\in W\\}$, where
$\langle\cdot,\cdot\rangle$ denotes the standard duality pairing of
$c_{00}\times c_{00}$. The space $X_{\mathcal{D}}$ is the completion of
$c_{00}$ under this norm. We fix a pair of lacunary sequences $m_{j}$,
$n_{j}$, $j\in\mathbb{N}$, of natural numbers. The first sequence is used to
define a collection of weights
$\\{\prod_{i=1}^{\ell}m_{j_{i}}:j_{1},\ldots,j_{\ell}\in\mathbb{N}\\}$, which
is closed under multiplication. To each weight $\prod_{i=1}^{\ell}m_{j_{i}}$,
we associate a Schreier family
$\mathcal{S}_{n_{j_{1}}+n_{j_{2}}+\cdots+n_{j_{\ell}}}$. This is a compact
collection of finite subsets of $\mathbb{N}$ recalled in Section 2. Every $f$
in $W$ satisfies $\|f\|_{\infty}\leq 1$ and it is either of the form $\pm
e_{i}^{*}$ or

(2) $f=\frac{1}{w(f)}\sum_{q=1}^{d}f_{q},$

where $w(f)=\prod_{i=1}^{\ell}m_{j_{i}}$ and $f_{1},\ldots,f_{d}$ are
successively supported vectors such that the set
$\\{\min\mathrm{supp}(f_{q}):1\leq q\leq d\\}$ is in the collection
$\mathcal{S}_{n_{j_{1}}+n_{j_{2}}+\cdots+n_{j_{d}}}$. A Banach space defined
by a norming set on which only the aforementioned restrictions are imposed is
asymptotic-$\ell_{1}$ (see, e.g., [1]), an asymptotic property implying having
a unique $\ell_{1}$ asymptotic model and thus also the Lebesgue property. To
avoid this without compromising the uniform uniqueness of $\ell_{1}$ spreading
models, just as in [5], we additionally demand that for $f$ in $W$ as in (2),
the sequence $w(f_{1}),\ldots,w(f_{d})$ is increasing sufficiently rapidly in
relation to the supports of $f_{1}$,…,$f_{d}$. Imposing these additional
restrictions on $W$ yields a Banach space with a uniformly unique $\ell_{1}$
spreading model and no asymptotic-$\ell_{1}$ subspace; however, as it was
proved in [11], it still satisfies the Lebesgue property. The intuitive
explanation for this is the following: despite the imposed restrictions on
$W$, there are sufficiently many functionals yielding the required
$\ell_{1}^{+}$ estimates on normalized collections
$(x_{\lambda})_{\lambda\in\mathcal{D}}$ in the defined Banach space.

To address this, we impose an additional metric constraint on $W$. Recall that
$\mathcal{D}$ is a totally bounded metric space with the metric
$d(\lambda,\mu)=2^{-n}$, where, if $\lambda\neq\mu$,
$n\in\mathbb{N}\cup\\{0\\}$ is the maximum height for which there exists
$\nu\in\\{0,1\\}^{n}$ with $\nu\leq\lambda$ and $\nu\leq\mu$. We fix a
bijection $\phi:\\{m_{j}:j\in\mathbb{N}\\}\to\mathcal{D}$ and to every $f\in
W$ of weight $\prod_{i=1}^{\ell}m_{j_{i}}$ we associate the node
$\lambda=\phi(m(f))$, where $m(f)$ is the maximum of
$m_{j_{1}},\ldots,m_{j_{\ell}}$. We then impose on $W$ the restriction that
for every $f$ as in (2), the nodes $\phi(m(f_{1})),\ldots,\phi(m(f_{d}))$
approximate some $\lambda$ in the completion of $\mathcal{D}$ and the
quantification of the approximation depends on the tuple
$(n_{j_{1}},\ldots,n_{j_{\ell}})$ and the supports of the $f_{q}$, $1\leq
q\leq d$. Modulo some subtle but significant details, the construction of the
space $X_{\mathcal{D}}$ is complete.

Using a standard technique, in every block-subspace $Y$ of $X_{\mathcal{D}}$,
we build a collection of vectors $(x_{m_{j}})_{j\in\mathbb{N}}$ such that, for
every $j\in\mathbb{N}$, the norm of $x_{m_{j}}$ can only be effectively
estimated by functionals of weight $w(f)=m_{j}$, i.e., $x_{m_{j}}$ is an exact
vector. We then consider the $\mathcal{D}$-indexed collection
$(x_{\phi^{-1}(\lambda)})_{\lambda\in\mathcal{D}}$. This witnesses the failure
of the Lebesgue property in $Y$. Indeed, for $n\in\mathbb{N}$, denote
$\\{0,1\\}^{n}=\\{\lambda_{1},\ldots,\lambda_{2^{n}}\\}$ and let
$m_{j_{1}}$,…,$m_{j_{2^{n}}}$ such that
$\phi(m_{j_{1}})\geq\lambda_{1}$,…,$\phi(m_{j_{2^{n}}})\geq\lambda_{2^{n}}$.
Then the nodes $\phi(m_{j_{1}})$,…$\phi(m_{j_{2^{n}}})$ form a $2^{-n}$
separated set in $\mathcal{D}$. For every $f\in W$ as in (2),
$\phi(m(f_{1}))$,…,$\phi(m(f_{d}))$ approximate a $\lambda$ in the completion
of $\mathcal{D}$, and thus, roughly speaking, the components of $f$ only
contribute to the norm of very few of the $x_{m_{j_{i}}}$, $1\leq i\leq
2^{n}$, yielding the $\ell_{\infty}$-estimate (1).

Our paper is organized as follows. In Section 2, we recall the required
terminology surrounding Schreier families and special convex combinations. We
also recall the definition of a Haar-$\ell_{1}^{+}$ sequence and remark on the
formulation of the Lebesgue property in terms of normalized collections
$(x_{\lambda})_{\lambda\in\mathcal{D}}$. In Section 3, we define the space
$X_{\mathcal{D}}$ and prove the uniform uniqueness of $\ell_{1}$ spreading
models. In Section 4, we define an auxiliary Banach space
$X_{\mathcal{D}}^{\mathrm{aux},\delta}$ and prove $\ell_{\infty}$-type
estimates for linear combinations of “simple” vectors in this space. In
Section 5, we define and study rapidly increasing sequences (RISs) and the
basic inequality. The former comprise a special class of sequences in
$X_{\mathcal{D}}$, and the latter provides bounds for their linear
combinations in terms of linear combinations of basis vectors in the auxiliary
space. In Section 6, we combine all ingredients prepared in Sections 4 and 5
to construct in every infinite-dimensional closed subspace of
$X_{\mathcal{D}}$ a normalized collection
$(x_{\lambda})_{\lambda\in\mathcal{D}}$ witnessing the failure of the Lebesgue
property.

##  2\. Preliminary information

Let $A,B\subset\mathbb{N}$. If $\max(A)<\min(B)$, then we write $A<B$. We also
write $n\leq A$ if $n\in\mathbb{N}$ is at most $\min(A)$ and, by convention,
$\emptyset<A$ and $A<\emptyset$ for every $A\subset\mathbb{N}$. We denote by
$c_{00}$ the vector space that consists of sequences of real numbers that have
at most finitely-many non-zero terms and we let $(e_{i})$ be the unit vector
basis of $c_{00}$. Note that the members of the norming set $W$ are themselves
members of $c_{00}$ and, in particular, that $\\{\pm
e_{i}\\}_{i=1}^{\infty}\subset W$. For this reason, we write $e^{*}_{i}\in
c_{00}$ when referring to $e_{i}$ as a norming functional. The support of
$x=(c_{i})\in c_{00}$ is the set $\text{supp}(x)=\\{i\in\mathbb{N}\mid
c_{i}\neq 0\\}$ and we say that vectors $x_{1},\ldots,x_{n}\in c_{00}$ are
successive if $\text{supp}(x_{i-1})<\text{supp}(x_{i})$ for $2\leq i\leq n$.
In this case, we use the shorthand notation $x_{1}<\ldots<x_{n}$ and we also
write $n\leq x$ if $n\leq\min\text{supp}(x)$. A finite or infinite sequence of
successive vectors in $c_{00}$ is said to be a block sequence of $(e_{i})$.

The Schreier sets are an increasing sequence of sets of finite subsets of
$\mathbb{N}$. These sets are defined inductively as follows:

$S_{0}=\\{\\{i\\}\mid i\in\mathbb{N}\\}\text{ and
}S_{1}=\\{F\subset\mathbb{N}\mid\text{card}(F)\leq F\\}$

and we write that

$S_{n+1}=\\{F\subset\mathbb{N}\mid F=\cup_{i=1}^{d}F_{i}\text{ where }d\leq
F_{1}<\ldots<F_{d}\in S_{n}\\}$

once $S_{n}$ has been defined. We also define the set
$\mathcal{A}_{3}=\\{F\subset\mathbb{N}\mid\text{card}(F)\leq 3\\}$ and we will
be interested in sets of the form

$S_{n}\ast\mathcal{A}_{3}=\\{F\subset\mathbb{N}\mid
F=\cup_{i=1}^{d}F_{i}\text{ where }F_{1}<\ldots<F_{d}\in\mathcal{A}_{3}\text{
and }\\{\min(F_{i})\\}_{i=1}^{d}\in S_{n}\\}\\}$

and it is easy to see that $F\in S_{n}\ast\mathcal{A}_{3}$ is the union of at
most three sets in $S_{n}$. A block sequence $x_{1}<\ldots<x_{k}\in c_{00}$ is
said to be $S_{n}$-admissible (resp. $S_{n}\ast\mathcal{A}_{3}$-admissible) if
$\\{\min\text{supp}(x_{i})\\}_{i=1}^{k}\in S_{n}$ (resp.
$\\{\min\text{supp}(x_{i})\\}_{i=1}^{k}\in S_{n}\ast\mathcal{A}_{3}$).

Let $n\in\mathbb{N}$ and let $\varepsilon>0$. Then, the vector $x=\sum_{i\in
F}c_{i}e_{i}\in c_{00}$ is said to be an $(n,\varepsilon)$-basic special
convex combination ($(n,\varepsilon)$-basic scc) if

  1. (1)

$F\in S_{n}$, $c_{i}\geq 0$ for each $i\in F$, and $\sum_{i\in F}c_{i}=1$

  2. (2)

For every $G\subset F$ with $G\in S_{n-1}$, we have that $\sum_{i\in
G}c_{i}<\varepsilon$

and we refer the reader to [1], [3], or [6] for more information. In
particular, note that $\sum_{i\in G}c_{i}<3\varepsilon$ if $G\subset F$ with
$G\in S_{k}\ast\mathcal{A}_{3}$ for $k<n$, and there is also the following
useful result which is Proposition 2.3 of [6].

######  Proposition 2.1.

Let $n\in\mathbb{N}$. Then, for every infinite subset $M\subset\mathbb{N}$ and
for every $F\subset M$ such that $F\in S_{n}$ is maximal with respect to
inclusion, there exist real numbers $(c_{i})_{i\in F}$ so that the vector
$x=\sum_{i\in F}c_{i}e_{i}\in c_{00}$ is an $(n,3/\min(F))$-basic scc.

Similarly, if $x_{1}<\ldots<x_{d}\in c_{00}$ and if
$s_{i}=\min\text{supp}(x_{i})$ for each $1\leq i\leq d$, then the vector
$x=\sum_{i=1}^{d}c_{i}x_{i}$ is said to be an $(n,\varepsilon)$-special convex
combination ($(n,\varepsilon)$-scc) if $\sum_{i=1}^{d}c_{i}e_{s_{i}}$ is an
$(n,\varepsilon)$-basic scc. It follows in this case that
$\sum_{i=1}^{d}c_{i}e_{t_{i}}$ is an $(n,2\varepsilon)$-basic scc as well
where $t_{i}=\max\text{supp}(x_{i})$ for each $1\leq i\leq d$ because

$\sum_{i\in G}c_{i}=\sum_{i\in G_{1}}c_{i}+\sum_{i\in
G_{2}}c_{i}<\varepsilon+\varepsilon=2\varepsilon$

if $G\subset\\{t_{i}\\}_{i=1}^{d}$ is $S_{n-1}$-admissible and is therefore a
translation to the right of at most two sets
$G_{1},G_{2}\subset\\{s_{i}\\}_{i=1}^{d}$ with $G_{1},G_{2}\in S_{n-1}$.

Finally, two specific asymptotic structures appear in this paper. The first is
that of a spreading model from [9], recalled in page 1. It is a consequence of
Ramsey’s theorem that every normalized basic sequence $(x_{i})$ in a Banach
space $X$ has a subsequence that generates a spreading model of $X$. We now
define the notion of a Haar-$\ell_{1}^{+}$ sequence and we need first the
definition of a Haar system of partitions of $\mathbb{N}$.

######  Definition 2.2.

A collection $(A^{n}_{j})_{j=0,n\in\mathbb{N}}^{2^{n}-1}$ of infinite subsets
of $\mathbb{N}$ is said to be a Haar system of partitions of $\mathbb{N}$ if
the following two conditions are satisfied:

  1. (1)

For every $n\in\mathbb{N}$, we have that
$\cup_{j=0}^{2^{n}-1}A^{n}_{j}=\mathbb{N}$ and $A^{n}_{j}\cap
A^{n}_{j^{\prime}}=\emptyset$ if $j\neq j^{\prime}$.

  2. (2)

For every $n\in\mathbb{N}$ and for every $0\leq j\leq 2^{n}-1$, we have that
$A^{n}_{j}=A^{n+1}_{2j}\cup A^{n+1}_{2j+1}$.

The connection between the Haar-$\ell_{1}^{+}$ condition (defined below) and
Riemann integrability is due to the fact that the infinite set $A^{n}_{j}$ of
the Haar system of partitions of $\mathbb{N}$ corresponds to the dyadic
rational numbers in the dyadic interval
$\left(\frac{j}{2^{n}},\frac{j+1}{2^{n}}\right)$ (see, e.g., [11, Theorem
3.1]).

######  Definition 2.3.

A normalized basic sequence $(x_{i})$ in $X$ is said to be Haar-$\ell_{1}^{+}$
if, for every Haar system of partitions
$(A^{n}_{j})_{j=0,n\in\mathbb{N}}^{2^{n}-1}$ of $\mathbb{N}$, there exists a
constant $C>0$ such that

$C\leq\lim_{n\to\infty}\sup\left\\{\frac{1}{2^{m}}\left\|\sum_{j=0}^{2^{m}-1}x_{i_{j}}\right\|\;\middle|\;m\geq
n\text{ and }2^{m}\leq i_{j}\in A^{m}_{j}\right\\}.$

######  Remark 2.4.

For a Banach space $X$ the following are equivalent.

  1. (1)

$X$ has the Lebesgue property.

  2. (2)

Every normalized basic sequence in $X$ is Haar-$\ell_{1}^{+}$.

  3. (3)

For every collection $(x_{\lambda})_{\lambda\in\mathcal{D}}$ of norm-one
vectors in $X$, there exists a constant $\theta>0$ such that, for every
$n\in\mathbb{N}$ and for every $\lambda\in\\{0,1\\}^{n}$, we have that

$\frac{1}{2^{n}}\left\|\sum_{\lambda\in\\{0,1\\}^{n}}x_{\mu_{\lambda}}\right\|\geq\theta$

for some collection of nodes $(\mu_{\lambda})_{\lambda\in\\{0,1\\}^{n}}$ with
$\mu_{\lambda}\geq\lambda$.

Indeed, the assertion $(1)\iff(2)$ is established in [11]. In addition, it is
easy to see that $(1)\implies(3)$ as collection witnessing the negation of (3)
can be easily used to construct a Dirichlet-type function with Riemann
integral zero. Lastly, for $(3)\implies(2)$, if there exists a normalized
basic sequence $(x_{i})$ in $X$ that is not Haar-$\ell_{1}^{+}$, then it is
easy to construct a collection $(x_{\lambda})_{\lambda\in\mathcal{D}}$ of
norm-one vectors for which the claim of $(3)$ does not hold as follows. Fix an
order preserving bijection $\mathcal{D}\to\\{A^{n}_{j}\mid
n\in\mathbb{N},0\leq j\leq 2^{n}-1\\}$ such that, for every $n\in\mathbb{N}$,
$\lambda\mapsto A^{|\lambda|}_{j_{\lambda}}$, for some $0\leq j_{\lambda}\leq
2^{|\lambda|}-1$. Then, for every $\lambda\in\mathcal{D}$, choose
$i_{\lambda}\in A^{|\lambda|}_{j_{\lambda}}$ such that $i_{\lambda}\geq
2^{|\lambda|}$ and put $x_{\lambda}=x_{i_{\lambda}}$. Note that
$(x_{\lambda})_{\lambda\in\mathcal{D}}$ can be chosen Schauder basic in the
lexicographical oder of $\mathcal{D}$.

##  3\. Construction of $X_{\mathcal{D}}$ and uniformly unique $\ell_{1}$
spreading model

In this section we define the norming set $W$ inducing the Banach space
$X_{\mathcal{D}}$. It is defined as an increasing union
$W=\cup_{k=0}^{\infty}W_{k}$, where $W_{0}=\\{\pm e_{i}^{*}\\}_{i=1}^{\infty}$
and, in each step, having defined $W_{k}$, we formulate the operations and
constraints that are used to produce $W_{k+1}$. The quantification of the
metric constraints discussed in the introduction is defined inductively and
carries a certain complexity; this is necessary to balance the uniform
uniqueness of $\ell_{1}$ spreading models with the $\ell_{\infty}$ estimates
on tree-indexed collection. At the end of this section, we prove that every
normalized weakly null sequence in $X_{\mathcal{D}}$ has a subsequence
generating a spreading model that is 102-equivalent to the unit vector basis
of $\ell_{1}$.

Fix a pair of strictly increasing sequences $(m_{j})$ and $(n_{j})$ of
positive integers such that $m_{1}=100$, $n_{1}=1$,

  1. (1)

$m_{j+1}\geq j^{2}m_{j}^{2}$, for every $j\in\mathbb{N}$,

  2. (2)

$n_{j+1}>\frac{4}{\log(100)}n_{j}\log(m_{j+1})$.

Let $\mathcal{D}=\cup_{n=0}^{\infty}\\{0,1\\}^{n}$ with the partial order and
metric $d$ defined in the introduction. We note that the completion
$\overline{\mathcal{D}}$ of $\mathcal{D}$ is a compact metric space. Denote
$\mathcal{M}=\\{\infty\\}\cup\\{m_{j}:j\in\mathbb{N}\\}$ and let
$\phi:\mathcal{M}\to\mathcal{D}$ be a bijection so that the nodes of
$\mathcal{D}$ are ordered lexicographically with respect to $\mathcal{M}$,
that is, $\phi(\infty)=\emptyset$, $\phi(m_{1})=(0)$, $\phi(m_{2})=(1)$,
$\phi(m_{3})=(0,0)$, and so on. Lastly, as a matter of notational convenience,
we always write $d(m_{j},\cdot)$ instead of the formally correct
$d(\phi(m_{j}),\cdot)$.

Now, let $W_{0}=\\{\pm e_{i}^{*}\\}_{i=1}^{\infty}\subset c_{00}$ and write
$w(\pm e^{*}_{i})=m(\pm e^{*}_{i})=\infty$ for each $i\in\mathbb{N}$. Then,
define the set

$W_{1}=W_{0}\cup\left\\{\frac{1}{m_{j_{1}}\cdots
m_{j_{l}}}\sum_{k=1}^{d}e^{*}_{i_{k}}\;\middle|\;\begin{subarray}{c}e^{*}_{i_{1}}<\ldots<e^{*}_{i_{d}}\text{
are }\\\
S_{n_{j_{1}}+\cdots+n_{j_{l}}}\text{-admissible}\end{subarray}\right\\}$

where $w(f)=m_{j_{1}}\cdots m_{j_{l}}$,
$\vec{w}(f)=(m_{j_{1}},\ldots,m_{j_{\ell}})$, and $m(f)=\max_{1\leq z\leq
l}m_{j_{z}}$ for $f\in W_{1}\setminus W_{0}$. We next define the following
notion of proximity with respect to $\overline{\mathcal{D}}$ for functionals
in $W_{1}$.

######  Definition 3.1.

The functionals $f_{1}<\ldots<f_{d}$ in $W_{1}$ are said to be
$(n,\lambda)$-packed for some $\lambda\in\overline{\mathcal{D}}$ if the
following conditions are satisfied:

  1. (1)

$f_{1}<\ldots<f_{d}$ are $S_{n}$-admissible

  2. (2)

$w(f_{i})\geq 2^{\max\text{supp}(f_{i-1})}$ for $2\leq i\leq d$

  3. (3)

$d(m(f_{i}),\lambda)\leq 2^{-\max\text{supp}(f_{i-1})}$ for $2\leq i\leq d$

and if, in addition, we have that $\max_{1\leq i\leq
d}d(m(f_{i}),\lambda)\leq\eta$, then $f_{1}<\ldots<f_{d}$ are said to be
$(n,\lambda,\eta)$-packed.

This definition allows for the following inductive definition as well.

######  Definition 3.2.

The functionals $f_{1}<\ldots<f_{d}$ in $W_{1}$ are said to be
$(n_{j_{1}},\ldots,n_{j_{l}},\lambda)$-packed for some
$\lambda\in\overline{\mathcal{D}}$ if the following claims hold:

  1. (1)

$w(f_{i})\geq 2^{\max\text{supp}(f_{i-1})}$ for $2\leq i\leq d$

  2. (2)

There exists a partition $F_{1}<\ldots<F_{M}$ of $\\{1,\ldots,d\\}$ such that
the set $\\{f_{\min(F_{k})}\\}_{k=1}^{M}$ is $S_{n_{j_{l}}}$-admissible

  3. (3)

There exist nodes $\lambda_{1},\ldots,\lambda_{M}\in\overline{\mathcal{D}}$
such that:

    1. (a)

$(f_{i})_{i\in F_{1}}$ is $(n_{j_{1}},\ldots,n_{j_{l-1}},\lambda_{1})$-packed

    2. (b)

$(f_{i})_{i\in F_{k}}$ is
$(n_{j_{1}},\ldots,n_{j_{l-1}},\lambda_{k},2^{-\max\text{supp}(f_{\max(F_{k-1})})})$-packed
for $2\leq k\leq M$

    3. (c)

$d(\lambda_{k},\lambda)\leq 2^{-\max\text{supp}(f_{\max(F_{k-1})})}$ for
$2\leq k\leq M$

and if, in addition, we have that $\max_{1\leq i\leq
d}d(m(f_{i}),\lambda)\leq\eta$, then we say that $f_{1}<\ldots<f_{d}$ are
$(n_{j_{1}},\ldots,n_{j_{l}},\lambda,\eta)$-packed.

We now define the next set of weighted functionals

$W_{2}=W_{1}\cup\left\\{\frac{1}{m_{j_{1}}\cdots
m_{j_{l}}}\sum_{i=1}^{d}f_{i}\;\middle|\;\begin{subarray}{c}f_{1}<\ldots<f_{d}\text{
are in }W_{1}\text{ and are}\\\
(n_{j_{1}},\ldots,n_{j_{l}},\lambda)\text{-packed for some
}\lambda\in\overline{\mathcal{D}}\end{subarray}\right\\}$

with $w(f)=m_{j_{1}}\cdots m_{j_{l}}$,
$\vec{w}(f)=(m_{j_{1}},\ldots,m_{j_{\ell}})$, and $m(f)=\max_{1\leq z\leq
l}m_{j_{z}}$. Now, once we have defined the sets $W_{0},W_{1},\ldots,W_{k}$,
we replace $W_{1}$ in Definitions 3.1 and 3.2 with $W_{k}$ and we write that

$W_{k+1}=W_{k}\cup\left\\{\frac{1}{m_{j_{1}}\cdots
m_{j_{l}}}\sum_{i=1}^{d}f_{i}\;\middle|\;\begin{subarray}{c}f_{1}<\ldots<f_{d}\text{
are in }W_{k}\text{ and are}\\\
(n_{j_{1}},\ldots,n_{j_{l}},\lambda)\text{-packed for some
}\lambda\in\overline{\mathcal{D}}\end{subarray}\right\\}$

for each $k\in\mathbb{N}$. We then define our norming set
$W=\cup_{k=0}^{\infty}W_{k}$ and we let the Banach space $X_{\mathcal{D}}$ be
the completion of $c_{00}$ with respect to the norm that is given by
$\|x\|=\sup\\{f(x)\mid f\in W\\}$ for $x\in c_{00}$.

######  Remark 3.3.

By induction on $k$, where $W=\cup_{k=0}^{\infty}W_{k}$, it is easy to prove
the following.

  1. (1)

For every $(n_{j_{1}},\ldots,n_{j_{\ell}},\lambda)$-packed sequence
$f_{1}<\cdots<f_{d}$ in $W$,

$f=\frac{1}{m_{j_{1}}\cdots m_{j_{\ell}}}\sum_{i=1}^{d}f_{i}$

is in $W$.

Incorporating a nested induction on $\ell$, the following items can be shown.

  1. (2)

A subsequence of a $(n_{j_{1}},\ldots,n_{j_{\ell}},\lambda)$-packed sequence
$f_{1}<\cdots<f_{d}$ in $W$ is also
$(n_{j_{1}},\ldots,n_{j_{\ell}},\lambda)$-packed.

  2. (3)

If $f_{1}<\cdots<f_{d}$ is a $(n_{j_{1}},\ldots,n_{j_{\ell}},\lambda)$-packed
sequence in $W$ and $g_{1}<\cdots<g_{d}$ are in $c_{00}$ such that, for $1\leq
i\leq d$ and $s\in\mathrm{supp}(g_{i})$, we have
$|g_{i}(e_{s})|=|f_{i}(e_{s})|$ then $g_{1}$,…,$g_{d}$ are also in $W$ and
they are $(n_{j_{1}},\ldots,n_{j_{\ell}},\lambda)$-tightly packed.

In particular, (2) and (3) yield that $W$ is closed under restricting the
supports of its members to subsets and changing the signs of their scalar
coefficients, and thus, $(e_{i})$ is a 1-unconditional basis for
$X_{\mathcal{D}}$.

We now prove that every spreading model of $X_{\mathcal{D}}$ is uniformly
equivalent to the unit vector basis of $\ell_{1}$. In fact, the precise
statement of the following theorem is slightly stronger than this assertion;
it will be required in its full generality later, in the proof of Proposition
5.3.

######  Theorem 3.4.

Let $(x_{r})$ be a normalized block sequence of $(e_{i})$. Then, there exists
an infinite subset $\Omega\subset\mathbb{N}$ such that, for every
$j_{0}\in\mathbb{N}$, there is the estimate

$\left\|\sum_{r\in F}c_{r}x_{r}\right\|\geq
0.99\frac{1}{m_{j_{0}}}\sum_{r\in\Gamma}|c_{r}|$

for every subset $\Gamma\subset\Omega$ such that $(x_{r})_{r\in\Gamma}$ are
$S_{n_{j_{0}}}$-admissible and for every choice $(c_{r})_{r\in\Gamma}$ of
scalars.

###### Proof.

Choose for every $r\in\mathbb{N}$ a norming functional $f_{r}\in W$ so that
$f_{r}(x_{r})=1$ and so that $\text{supp}(f_{r})\subset\text{supp}(x_{r})$.
Then, there are two cases to consider.

  * (i)

$\limsup_{r\to\infty}w(f_{r})=L<\infty$

Here, we may pass to an infinite subset $\Omega_{1}\subset\mathbb{N}$ so that
$w(f_{r})=m_{j_{1}}\cdots m_{j_{l}}$ for each $r\in\Omega_{1}$. Note that each
$f_{r}$ with $r\in\Omega_{1}$ is then
$(n_{j_{1}},\ldots,n_{j_{l}},\lambda_{r})$-packed for some
$\lambda_{r}\in\overline{\mathcal{D}}$. We may then pass to a further infinite
subset $\Omega_{2}\subset\Omega_{1}$ so that
$d(\lambda_{r},\lambda)\leq\frac{1}{2}\cdot 2^{-\max\text{supp}(x_{r-1})}$ for
each $r\in\Omega_{2}\setminus\\{\min(\Omega_{2})\\}$ and for some
$\lambda\in\overline{\mathcal{D}}$. Now, let $\Omega=\Omega_{2}$ and consider
the subsequence $(x_{r})_{r\in\Omega_{2}}$. Let $\Gamma\subset\Omega$ be a
subset such that the block vectors $(x_{r})_{r\in\Gamma}$ are
$S_{n_{j_{0}}}$-admissible and let

$f_{r}=\frac{1}{m_{j_{1}}\cdots m_{j_{l}}}\sum_{q=1}^{d_{r}}f^{r}_{q}\qquad
r\in\Gamma$

be the associated norming functionals. For each $r\in\Gamma$, let
$F^{r}_{1}<\ldots<F^{r}_{M_{r}}$ be the partition of $\\{1,\ldots,d_{r}\\}$
and let $\lambda^{r}_{1},\ldots,\lambda^{r}_{M_{r}}\in\overline{\mathcal{D}}$
be the corresponding nodes so that Definition 3.2 is satisfied because the
functionals $f^{r}_{1}<\ldots<f^{r}_{d_{r}}$ are
$(n_{j_{1}},\ldots,n_{j_{l}},\lambda_{r})$-packed. We will now show that the
concatenation

$f^{\min(\Gamma)}_{\min(F^{\min(\Gamma)}_{2})}<\ldots<f^{\min(\Gamma)}_{d_{\min(\Gamma)}}<\ldots<f^{\max(\Gamma)}_{\min(F_{2}^{\max(\Gamma)})}<\ldots<f^{\max(\Gamma)}_{d_{\max(\Gamma)}}$

is in $W$. First, note that there is some $\kappa\in\mathbb{N}$ so that each
functional in this concatenation is a member of $W_{\kappa}$. It therefore
remains to show that these functionals are
$(n_{j_{1}},\ldots,n_{j_{l}},n_{j_{0}},\lambda)$-packed. Indeed, it follows
immediately that they are
$S_{n_{j_{1}}+\cdots+n_{j_{l}}+n_{j_{0}}}$-admissible and are very fast
growing in the sense of Definition 3.2. Then, let
$F_{\min(\Gamma)}<\ldots<F_{\max(\Gamma)}$ be the partition of the number of
functionals in the above concatenation such that $(f_{i})_{i\in
F_{r}}=\\{f^{r}_{\min(F^{r}_{2})},\ldots,f^{r}_{d_{r}}\\}$ and note that
$\\{f_{\min(F_{r})}\\}_{r\in\Gamma}=\\{f^{r}_{\min(F^{r}_{2})}\\}_{r\in\Gamma}$
is $S_{n_{j_{0}}}$-admissible. Moreover:

  1. (1)

$(f_{i})_{i\in F_{1}}=\\{f^{1}_{\min(F^{1}_{2})},\ldots,f^{1}_{d_{1}}\\}$ is
$(n_{j_{1}},\ldots,n_{j_{l}},\lambda_{1})$-packed.

  2. (2)

$(f_{i})_{i\in F_{r}}=\\{f^{r}_{\min(F^{r}_{2})},\ldots,f^{r}_{d_{r}}\\}$ is
$(n_{j_{1}},\ldots,n_{j_{l}},\lambda_{r})$-packed for each
$r\in\Gamma\setminus\\{\min(\Gamma)\\}$ and, in addition, for each $f_{i}$
with $i\in F^{r}_{k}$ (where $2\leq k\leq M_{r}$), there is

$\displaystyle d(m(f_{i}),\lambda_{r})$ $\displaystyle\leq
d(m(f_{i}),\lambda^{r}_{k})+d(\lambda^{r}_{k},\lambda_{r})$ $\displaystyle\leq
2\cdot 2^{-\max\text{supp}(f_{\max(F^{r}_{k-1})})}\leq
2^{-\max\text{supp}(f_{\max(F_{r-1})})}$

so that, in fact, $(f_{i})_{i\in F_{r}}$ is
$(n_{j_{1}},\ldots,n_{j_{l}},\lambda_{r},2^{-\max\text{supp}(f_{\max(F_{r-1})})})$-packed.

  3. (3)

$d(\lambda_{r},\lambda)\leq\frac{1}{2}\cdot 2^{-\max\text{supp}(x_{r-1})}\leq
2^{-\max\text{supp}(f_{\max(F_{r-1})})}$ for each
$r\in\Gamma\setminus\\{\min(\Gamma)\\}$.

Putting these pieces together, it follows that the functional

$\mathscr{F}=\frac{1}{m_{j_{1}}\cdots
m_{j_{l}}m_{j_{0}}}\sum_{r\in\Gamma}\sum_{q=\min(F^{r}_{2})}^{d_{r}}f^{r}_{q}$

is $(n_{j_{1}},\ldots,n_{j_{l}},n_{j_{0}},\lambda)$-packed and is therefore a
member of $W_{\kappa+1}\subset W$. Now, because $(x_{r})_{r\in\Omega}$ is
itself $1$-unconditional, there is the estimate

$\displaystyle\left\|\sum_{r\in\Gamma}c_{r}x_{r}\right\|=\left\|\sum_{r\in\Gamma}|c_{r}|x_{r}\right\|$
$\displaystyle\geq\mathscr{F}\left(\sum_{r\in\Gamma}|c_{r}|x_{r}\right)$
$\displaystyle=\frac{1}{m_{j_{1}}\cdots
m_{j_{l}}m_{j_{0}}}\sum_{r\in\Gamma}\sum_{q=\min(F^{r}_{2})}^{d_{r}}f^{r}_{q}\left(\sum_{p\in\Gamma}|c_{p}|x_{p}\right)$
$\displaystyle=\frac{1}{m_{j_{0}}}\sum_{r\in\Gamma}\frac{|c_{r}|}{m_{j_{1}}\cdots
m_{j_{l}}}\sum_{q=\min(F^{r}_{2})}^{d_{r}}f^{r}_{q}(x_{r})$
$\displaystyle=\frac{1}{m_{j_{0}}}\sum_{r\in\Gamma}|c_{r}|\left(f_{r}(x_{r})-\frac{1}{m_{j_{1}}\cdots
m_{j_{l}}}\sum_{q\in F^{r}_{1}}f^{r}_{q}(x_{r})\right)$
$\displaystyle\geq\frac{1}{m_{j_{0}}}\sum_{r\in\Gamma}|c_{r}|\left(1-\frac{1}{m_{j_{l}}}\cdot\frac{1}{m_{j_{1}}\cdots
m_{j_{l-1}}}\sum_{q\in F^{r}_{1}}f^{r}_{q}(x_{r})\right)$
$\displaystyle\geq\frac{1}{m_{j_{0}}}\sum_{r\in\Gamma}|c_{r}|\left(1-\frac{1}{m_{1}}\right)=\frac{m_{1}-1}{m_{1}m_{j_{0}}}\sum_{r\in\Gamma}|c_{r}|$

where we have used the fact that $\frac{1}{m_{j_{1}}\cdots
m_{j_{l-1}}}\sum_{q\in F^{r}_{1}}f^{r}_{q}\in W$, and this completes the proof
for the first case.

  * (ii)

$\limsup_{r\to\infty}w(f_{r})=\infty$

In this case, there exists an infinite subset $\Omega_{1}\subset\mathbb{N}$
such that $w(f_{r})$ for $r\in\Omega_{1}\setminus\\{\min(\Omega_{1})\\}$ is
very fast growing in the sense of Definition 3.1. Then, let $\lambda_{r}$ be
the node in $\overline{\mathcal{D}}$ that corresponds to $m(f_{r})$ for each
$r\in\Omega_{1}$. We may then pass to a further infinite subset
$\Omega_{2}\subset\Omega_{1}$ so that
$d(\lambda_{r},\lambda)\leq\frac{1}{2}\cdot 2^{-\max\text{supp}(x_{r-1})}$ for
some $\lambda\in\overline{\mathcal{D}}$ and for each
$r\in\Omega_{2}\setminus\\{\min(\Omega_{2})\\}$. Now, let $\Omega=\Omega_{2}$
and consider the subsequence $(x_{r})_{r\in\Omega}$. Let $\Gamma\subset\Omega$
be a subset such that the block vectors $(x_{r})_{r\in\Gamma}$ are
$S_{n_{j_{0}}}$-admissible and let $f_{\min(\Gamma)}<\ldots<f_{\max(\Gamma)}$
be the corresponding norming functionals. Note that there is some
$\kappa\in\mathbb{N}$ so that $f_{r}\in W_{\kappa}$ for each $r\in\Gamma$.
Note also that, by construction, these norming functionals are
$S_{n_{j_{0}}}$-admissible and are very fast growing in the sense of
Definition 3.1. Moreover, there is the estimate

$d(\lambda_{r},\lambda)\leq 2^{-\max\text{supp}(x_{r-1})}\leq
2^{-\max\text{supp}(f_{r-1})}$

for each $r\in\Gamma\setminus\\{\min(\Gamma)\\}$ so that these functionals are
$(n_{j_{0}},\lambda)$-packed. Then, we have that
$\frac{1}{m_{j_{0}}}\sum_{r\in\Gamma}f_{r}\in W_{\kappa+1}\subset W$ so that

$\displaystyle\left\|\sum_{r\in\Gamma}c_{r}x_{r}\right\|=\left\|\sum_{r\in\Gamma}|c_{r}|x_{r}\right\|$
$\displaystyle\geq\frac{1}{m_{j_{0}}}\sum_{r\in\Gamma}f_{r}\left(\sum_{p\in\Gamma}|c_{p}|x_{p}\right)$
$\displaystyle=\frac{1}{m_{j_{0}}}\sum_{r\in\Gamma}|c_{r}|f_{r}(x_{r})=\frac{1}{m_{j_{0}}}\sum_{r\in\Gamma}|c_{r}|\geq\frac{m_{1}-1}{m_{1}m_{j_{0}}}\sum_{r\in\Gamma}|c_{r}|$

and this completes the proof of the theorem. ∎

##  4\. Auxiliary Banach spaces and $c_{0}$-type estimate

So far we have proved that all spreading models admitted by normalized weakly
null sequences in $X_{\mathcal{D}}$ are uniformly equivalent to the unit
vector basis of $\ell_{1}$. The remaining goal is to define in each subspace
of $X_{\mathcal{D}}$ a normalized collection
$(x_{\lambda})_{\lambda\in\mathcal{D}}$ satisfying the
$\ell_{\infty}$-estimate (1), and this will require some effort. As a first
step, we define an auxiliary norming set $W^{\text{aux},\delta}$ inducing a
norm on $c_{00}$ and an auxiliary Banach space
$X_{\mathcal{D}}^{\text{aux},\delta}$. We then prove estimates for the norm of
linear combinations of “simple” vectors, specifically weighted
$(n,\varepsilon)$ basic special convex combinations, in the auxiliary space.
These will be relevant later when a tool called the basic inequality will
yield high-precision estimates of linear combinations of special sequences of
vectors, called rapidly increasing sequences, in $X_{\mathcal{D}}$, in terms
of sequences of basis vectors in $X_{\mathcal{D}}^{\text{aux},\delta}$.

We derive in this section a $c_{0}$-type estimate for the action of auxiliary
functionals on weighted $(n,\varepsilon)$-basic scc vectors. The idea is that
the functional evaluates to something small if its weight is in some sense a
mismatch to the $(n,\varepsilon)$-basic scc vector weight. Let us first define
the auxiliary functionals and auxiliary Banach spaces.

We will define an auxiliary Banach space for each $\delta>0$. First, fix
$\delta>0$ and let $W_{0}^{\text{aux},\delta}=\\{\pm
e_{i}^{*}\\}_{i=1}^{\infty}\subset c_{00}$ as before. Then, also as before,
define $w(\pm e^{*}_{i})=m(\pm e^{*}_{i})=\infty$ and define the set of
auxiliary weighted functionals

$W^{\text{aux},\delta}_{1}=W^{\text{aux},\delta}_{0}\cup\left\\{\frac{1}{m_{j_{1}}\cdots
m_{j_{l}}}\sum_{k=1}^{d}e^{*}_{i_{k}}\;\middle|\;\begin{subarray}{c}e^{*}_{i_{1}}<\ldots<e^{*}_{i_{d}}\text{
are }\\\
S_{n_{j_{1}}+\cdots+n_{j_{l}}}\ast\mathcal{A}_{3}\text{-admissible}\end{subarray}\right\\}$

where $w(f)=m_{j_{1}}\cdots m_{j_{l}}$, $m(f)=\max_{1\leq z\leq l}m_{j_{z}}$,
and $\vec{w}(f)=(m_{j_{1}},\ldots,m_{j_{\ell}})$, for $f\in
W_{1}^{\text{aux},\delta}\setminus W_{0}^{\text{aux},\delta}$. We next define
the following notion of proximity with respect to $\overline{\mathcal{D}}$ for
functionals in $W^{\text{aux},\delta}_{1}$.

######  Definition 4.1.

The functionals $f_{1}<\ldots<f_{d}$ in $W^{\text{aux},\delta}_{1}$ are said
to be $(n,\lambda)_{\delta}$-packed for some
$\lambda\in\overline{\mathcal{D}}$ if the following conditions are satisfied:

  1. (1)

$f_{1}<\ldots<f_{d}$ are $S_{n}\ast\mathcal{A}_{3}$-admissible

  2. (2)

$w(f_{i})\geq 2^{\max\text{supp}(f_{i-2})}$ for $3\leq i\leq d$

  3. (3)

$d(m(f_{i}),\lambda)\leq\delta$ for $f_{i}\notin W^{\text{aux},\delta}_{0}$
and for $2\leq i\leq d$

This definition is the base case for the following inductive definition.

######  Definition 4.2.

The functionals $f_{1}<\ldots<f_{d}$ in $W^{\text{aux},\delta}_{1}$ are said
to be $(n_{j_{1}},\ldots,n_{j_{l}},\lambda)_{\delta}$-packed for some
$\lambda\in\overline{\mathcal{D}}$ if the following conditions hold:

  1. (1)

$w(f_{i})\geq 2^{\max\text{supp}(f_{i-2})}$ for $3\leq i\leq d$

  2. (2)

There exists a partition $F_{1}<\ldots<F_{M}$ of $\\{1,\ldots,d\\}$ such that
the set $\\{f_{\min(F_{k})}\\}_{k=1}^{M}$ is $S_{n_{j_{l}}}$-admissible and
such that the functionals $(f_{i})_{i\in F_{k}}$ are
$S_{n_{j_{1}}+\cdots+n_{j_{l-1}}}\ast\mathcal{A}_{3}$-admissible for $1\leq
k\leq M$

  3. (3)

There exists a node $\lambda_{1}\in\overline{\mathcal{D}}$ such that
$(f_{i})_{i\in F_{1}}$ is
$(n_{j_{1}},\ldots,n_{j_{l-1}},\lambda_{1})_{\delta}$-packed

  4. (4)

$d(m(f_{i}),\lambda)\leq\delta$ for
$f_{i}\in\left(\bigcup_{k=2}^{M}F_{k}\right)\setminus
W^{\text{aux},\delta}_{0}$

It is worth noting that a subset of
$(n_{j_{1}},\ldots,n_{j_{l}},\lambda)_{\delta}$-packed functionals in
$W^{\text{aux},\delta}_{1}$ is still
$(n_{j_{1}},\ldots,n_{j_{l}},\lambda)_{\delta}$-packed, and we now define the
next set of auxiliary weighted functionals

$W^{\text{aux},\delta}_{2}=W^{\text{aux},\delta}_{1}\cup\left\\{\frac{1}{m_{j_{1}}\cdots
m_{j_{l}}}\sum_{i=1}^{d}f_{i}\;\middle|\;\begin{subarray}{c}f_{1}<\ldots<f_{d}\text{
are in }W^{\text{aux},\delta}_{1}\text{ and are}\\\
(n_{j_{1}},\ldots,n_{j_{l}},\lambda)_{\delta}\text{-packed for some
}\lambda\in\overline{\mathcal{D}}\end{subarray}\right\\}$

with $w(f)=m_{j_{1}}\cdots m_{j_{l}}$, $m(f)=\max_{1\leq z\leq l}m_{j_{z}}$,
and $\vec{w}(f)=(m_{j_{1}},\ldots,m_{j_{\ell}})$. Now, once we have defined
the sets
$W^{\text{aux},\delta}_{0},W^{\text{aux},\delta}_{1},\ldots,W^{\text{aux},\delta}_{k}$,
we replace $W^{\text{aux},\delta}_{1}$ in Definitions 4.1 and 4.2 with
$W^{\text{aux},\delta}_{k}$ and we write that

$W^{\text{aux},\delta}_{k+1}=W^{\text{aux},\delta}_{k}\cup\left\\{\frac{1}{m_{j_{1}}\cdots
m_{j_{l}}}\sum_{i=1}^{d}f_{i}\;\middle|\;\begin{subarray}{c}f_{1}<\ldots<f_{d}\text{
are in }W^{\text{aux},\delta}_{k}\text{ and are}\\\
(n_{j_{1}},\ldots,n_{j_{l}},\lambda)_{\delta}\text{-packed for some
}\lambda\in\overline{\mathcal{D}}\end{subarray}\right\\}$

for each $k\in\mathbb{N}$. We then define our norming set
$W^{\text{aux},\delta}=\bigcup_{k=0}^{\infty}W^{\text{aux},\delta}_{k}$ and we
let the Banach space $X_{\mathcal{D}}^{\text{aux},\delta}$ be the completion
of $c_{00}$ with respect to the norm that is given by
$\|x\|_{W^{\text{aux},\delta}}=\sup\\{f(x)\mid f\in W^{\text{aux},\delta}\\}$
for $x\in c_{00}$. We will need the following lemma before we can prove the
main result of this section.

######  Lemma 4.3.

Let $f=\frac{1}{m_{j_{1}}\cdots m_{j_{l}}}\sum_{i=1}^{d}f_{i}\in
W^{\text{aux},\delta}$. Then, $\\{i\mid f_{i}\notin
W^{\text{aux},\delta}_{0}\\}=E_{0}\cup E_{1}\cup\ldots\cup E_{l}$ where
$\mathrm{card}(E_{0})\leq 1$ and, for each $1\leq z\leq l$, $(f_{i})_{i\in
E_{z}}$ is $S_{n_{j_{1}}+\cdots+n_{j_{z}}}\ast\mathcal{A}_{3}$-admissible and
$\max_{i\in E_{z}}d_{T}(m(f_{i}),\lambda_{z})\leq\delta$ for some
$\lambda_{z}\in\overline{\mathcal{D}}$.

###### Proof.

We proceed by induction on the length $l$ of the tuple
$(m_{j_{1}},\ldots,m_{j_{l}})$. If $l=1$, then
$f=\frac{1}{m_{j_{1}}}\sum_{i=1}^{d}f_{i}\in W^{\text{aux},\delta}_{k}$ for
some $k\geq 1$ and we put

$\displaystyle E_{0}$ $\displaystyle=\\{1\\}\setminus\\{i\mid f_{i}\in
W^{\text{aux},\delta}_{0}\\}$ $\displaystyle E_{1}$
$\displaystyle=\\{2,\ldots,d\\}\setminus\\{i\mid f_{i}\in
W^{\text{aux},\delta}_{0}\\}$

and note that, if $k=1$, then $E_{0}=E_{1}=\emptyset$ so that the result holds
vacuously. Assume then that $k>1$ and note that $f_{1}<\ldots<f_{d}$ are in
$W^{\text{aux},\delta}_{k-1}$ and are $(n_{j_{1}},\lambda)_{\delta}$-packed
for some $\lambda\in\overline{\mathcal{D}}$. It therefore follows by
definition that $(f_{i})_{i\in E_{1}}$ are
$S_{n_{j_{1}}}\ast\mathcal{A}_{3}$-admissible and that $\max_{i\in
E_{1}}d(m(f_{i}),\lambda)\leq\delta$.

Assume now that the result holds for functionals of the form
$f=\frac{1}{m_{j_{1}}\cdots m_{j_{l}}}\sum_{i=1}^{d}f_{i}\in
W^{\text{aux},\delta}$ and let $g=\frac{1}{m_{j_{1}}\cdots
m_{j_{l}}m_{j_{l+1}}}\sum_{i=1}^{d}g_{i}\in W^{\text{aux},\delta}$. As above,
the result holds vacuously if $g\in W^{\text{aux},\delta}_{1}$, so assume that
$g\in W^{\text{aux},\delta}_{k}$ for some $k>1$. In this case, we have that
$g_{1}<\ldots<g_{d}$ are in $W^{\text{aux},\delta}_{k-1}$ and are
$(n_{j_{1}},\ldots,n_{j_{l}},n_{j_{l+1}},\lambda)_{\delta}$-packed for some
$\lambda\in\overline{\mathcal{D}}$. It follows that there is a partition
$F_{1}<\ldots<F_{M}$ of $\\{1,\ldots,d\\}$ as in Definition 4.2. Note that in
particular that $(g_{i})_{i\in F_{1}}$ are
$(n_{j_{1}},\ldots,n_{j_{l}},\lambda_{1})_{\delta}$-packed for some
$\lambda_{1}\in\overline{\mathcal{D}}$ so that the functional
$\tilde{g}=\frac{1}{m_{j_{1}}\cdots m_{j_{l}}}\sum_{i\in F_{1}}g_{i}\in
W^{\text{aux},\delta}$. Then, by the inductive hypothesis, there are sets
$E_{0}\cup E_{1}\cup\ldots\cup E_{l}\subset F_{1}$ so that the required claim
holds for $\tilde{g}$. Now, let
$E_{l+1}=\left(\bigcup_{r=2}^{M}F_{r}\right)\setminus\\{i\mid g_{i}\in
W^{\text{aux},\delta}_{0}\\}$ and it follows that the required claim holds for
$g$ itself so that the proof is complete. ∎

We now derive the aforementioned $c_{0}$-type estimate for auxiliary weighted
functionals whose weights in some sense mismatch the weights of weighted
$(n,\varepsilon)$-basic scc vectors. First, we make a remark about the
relation between the weight of a functional and the admissibility of its
components.

######  Remark 4.4.

Let $j\in\mathbb{N}$ and $f\in W$ such $m(f)<m_{j}$ and $w(f)<m_{j}^{4}$,
then, if $\vec{w}(f)=(m_{j_{1}},\ldots,m_{j_{\ell}})$,
$n_{j_{1}}+\cdots+n_{j_{\ell}}<n_{j}$. Indeed, $m_{1}^{l}\leq w(f)<m_{j}^{4}$,
and thus, $l<4\log(m_{j})/\log(100)$. By condition (2) on page (2),

$n_{j_{1}}+\cdots+n_{j_{l}}\leq
ln_{j-1}<n_{j-1}4\frac{\log(m_{j})}{\log(100)}\leq n_{j}.$

######  Proposition 4.5.

Let $N\in\mathbb{N}\cup\\{0\\}$ and choose $(t_{i})_{i=1}^{2^{N}}$ such that
$d(m_{t_{i}},m_{t_{i^{\prime}}})\geq 2^{-N}$ for each $i\neq i^{\prime}$.
Then, choose $\varepsilon_{i}\in\left(0,\frac{1}{3m_{t_{i}}2^{N}}\right)$ and
define a $(n_{t_{i}},\varepsilon_{i})$-basic scc by

$x_{i}=m_{t_{i}}\sum_{r\in F_{i}}c_{r}^{i}e_{i_{r}}$

for each $1\leq i\leq 2^{N}$. It follows that
$\left|f\left(\sum_{i\in\Omega}a_{i}x_{i}\right)\right|\leq\left(1+\frac{4}{\sqrt{w(f)}}\right)\max_{i\in\Omega}|a_{i}|$
for every subset $\Omega\subset\\{1,\ldots,2^{N}\\}$, for every choice
$(a_{i})_{i\in\Omega}$ of scalars, and for every auxiliary weighted functional
$f=\frac{1}{m_{j_{1}}\cdots m_{j_{l}}}\sum_{q=1}^{d}f_{q}\in
W^{\text{aux},\delta}$ where $\delta<2^{-(N+1)}$.

###### Proof.

Let $\Omega\subset\\{1,\ldots,2^{N}\\}$ be given. We will proceed by induction
on the level $k$ norming sets $W^{\text{aux},\delta}_{k}$ and the inductive
hypothesis will include that $\left|f\left(\sum_{i\in
A_{f}(\Omega)}a_{i}x_{i}\right)\right|<\frac{4\gamma}{\sqrt{w(f)}}$ if
$f\notin W_{0}^{\text{aux},\delta}$ where $\gamma=\max_{i\in\Omega}|a_{i}|$
and where $A_{f}(\Omega):=\\{i\in\Omega\mid m_{t_{i}}\neq m(f)\\}$. The base
case is the level $k=1$ norming set and we first let $f\in
W^{\text{aux},\delta}_{0}$ so that

$\left|f\left(\sum_{i\in\Omega}a_{i}x_{i}\right)\right|=\left|e^{*}_{i_{0}}\left(\sum_{i\in\Omega}a_{i}m_{t_{i}}\sum_{r\in
F_{i}}c^{i}_{r}e_{r}\right)\right|\leq\max_{i\in\Omega}\varepsilon_{i}m_{t_{i}}|a_{i}|<\max_{i\in\Omega}|a_{i}|$

for some $i_{0}\in\mathbb{N}$. It therefore remains to check functionals of
the form

$f=\frac{1}{m_{j_{1}}\cdots m_{j_{l}}}\sum_{q=1}^{d}e^{*}_{i_{q}}\in
W^{\text{aux},\delta}_{1}\setminus W^{\text{aux},\delta}_{0}$

and we write $A_{f}(\Omega)=\\{i\in\Omega\mid m_{t_{i}}\neq m(f)\\}=A_{1}\cup
A_{2}\cup A_{3}$ where

$\displaystyle A_{1}$ $\displaystyle=\\{i\in\Omega\mid m_{t_{i}}<m(f)\\}$
$\displaystyle A_{2}$ $\displaystyle=\\{i\in\Omega\mid m(f)<m_{t_{i}}\text{
and }n_{j_{1}}+\cdots+n_{j_{l}}<n_{t_{i}}\\}$ $\displaystyle A_{3}$
$\displaystyle=\\{i\in\Omega\mid m(f)<m_{t_{i}}\text{ and
}n_{j_{1}}+\cdots+n_{j_{l}}\geq n_{t_{i}}\\}$

so that $\Omega\setminus A_{f}(\Omega)$ is at most a singleton $\\{i_{0}\\}$
with $m_{t_{i_{0}}}=m(f)$. Then, we have that

$\left|f\left(\sum_{i\in\Omega}a_{i}x_{i}\right)\right|\leq\sum_{p=1}^{3}\left|f\left(\sum_{i\in
A_{p}}a_{i}x_{i}\right)\right|+|f(a_{i_{0}}x_{i_{0}})|\leq\sum_{p=1}^{3}\left|f\left(\sum_{i\in
A_{p}}a_{i}x_{i}\right)\right|+\max_{i\in\Omega}|a_{i}|$

because $|f(x_{i_{0}})|\leq\frac{m_{t_{i_{0}}}}{w(f)}\leq 1$. Now, recall that
$\gamma=\max_{i\in\Omega}|a_{i}|$ and note that, for $A_{1}$, there is the
estimate

$\displaystyle\left|f\left(\sum_{i\in A_{1}}a_{i}x_{i}\right)\right|$
$\displaystyle\leq\sum_{i\in
A_{1}}|f(a_{i}x_{i})|\leq\frac{\gamma}{w(f)}\sum_{i\in A_{1}}m_{t_{i}}$
$\displaystyle\leq\frac{\gamma}{w(f)}\text{card}(A_{1})m_{t_{\max(A_{1})}}\leq\frac{\gamma}{w(f)}t_{\max(A_{1})}m_{t_{\max(A_{1})}}$
$\displaystyle\leq\frac{\gamma}{w(f)}(\text{ind}(m(f))-1)m_{\text{ind}(m(f))-1}\leq\frac{\gamma
m_{\text{ind}(m(f))}}{w(f)\sqrt{m_{\text{ind}(m(f))}}}\leq\frac{\gamma}{\sqrt{w(f)}}$

where we have used the first condition on the sequence $(m_{j})$ and the fact
that $t_{i}<\text{ind}(m(f)):=\max_{1\leq z\leq l}j_{z}$ for each $i\in
A_{1}$. Next, for $A_{2}$, note that

$\displaystyle\left|f\left(\sum_{i\in A_{2}}a_{i}x_{i}\right)\right|$
$\displaystyle\leq\frac{\gamma}{w(f)}\sum_{i\in
A_{2}}m_{t_{i}}\left|\sum_{q=1}^{d}e^{*}_{i_{q}}\left(\sum_{r\in
F_{i}}c^{i}_{r}e_{r}\right)\right|$
$\displaystyle\leq\frac{\gamma}{w(f)}\sum_{i\in
A_{2}}3m_{t_{i}}\varepsilon_{i}\leq\frac{\gamma}{w(f)}\sum_{i\in
A_{2}}\frac{1}{2^{N}}\leq\frac{\gamma}{w(f)}\leq\frac{\gamma}{\sqrt{w(f)}}$

where we have used the observation that immediately precedes Proposition 2.1
and the fact that $e^{*}_{i_{1}}<\ldots<e^{*}_{i_{d}}$ are
$S_{n_{j_{1}}+\cdots+n_{j_{l}}}\ast\mathcal{A}_{3}$-admissible with
$n_{j_{1}}+\cdots+n_{j_{l}}<n_{t_{i}}$ for each $i\in A_{2}$. Lastly, for
$A_{3}$, note that

$\displaystyle\left|f\left(\sum_{i\in A_{3}}a_{i}x_{i}\right)\right|$
$\displaystyle\leq\frac{\gamma}{w(f)}\sum_{i\in
A_{3}}m_{t_{i}}\leq\frac{\gamma}{w(f)}|A_{3}|m_{t_{\max(A_{3})}}$
$\displaystyle\leq\frac{\gamma
m^{2}_{t_{\max(A_{3})}}}{w(f)}\leq\frac{\gamma\sqrt{w(f)}}{w(f)}=\frac{\gamma}{\sqrt{w(f)}}$

because $m_{t_{i}}^{2}\leq\sqrt{w(f)}$ for each $i\in A_{3}$, by Remark 4.4.
It now follows by putting these pieces together that

$\left|f\left(\sum_{i\in\Omega}a_{i}x_{i}\right)\right|\leq\gamma+\frac{3\gamma}{\sqrt{w(f)}}$

for each $f\in W^{\text{aux},\delta}_{1}$ and, in particular, we have that
$\left|f\left(\sum_{i\in
A_{f}(\Omega)}a_{i}x_{i}\right)\right|<\frac{4\gamma}{\sqrt{w(f)}}$ as is
required as part of the inductive hypothesis.

Assume now that the statement of Proposition 4.5 holds for every functional
$f\in W^{\text{aux},\delta}_{k}$, let $\Omega\subset\\{1,\ldots,2^{N}\\}$ be
given, and let

$f=\frac{1}{m_{j_{1}}\cdots m_{j_{l}}}\sum_{q=1}^{d}f_{q}\in
W^{\text{aux},\delta}_{k+1}\setminus W^{\text{aux},\delta}_{k}$

where $f_{q}\in W^{\text{aux},\delta}_{k}$ for each $1\leq q\leq d$. Define
the sets $B=\\{q\mid f_{q}\in W^{\text{aux},\delta}_{0}\\}$ and
$C=\\{3,\ldots,d\\}\setminus B$. We may then write
$f=\sum_{\alpha=1}^{4}g_{\alpha}$ where

$g_{1}=\frac{1}{w(f)}f_{1},\text{ }g_{2}=\frac{1}{w(f)}f_{2},\text{
}g_{3}=\frac{1}{w(f)}\sum_{q\in B}f_{q},\text{ and
}g_{4}=\frac{1}{w(f)}\sum_{q\in C}f_{q}$

and it follows that

$\left|f\left(\sum_{i\in\Omega}a_{i}x_{i}\right)\right|\leq\left|f\left(\sum_{i\in
A_{f}(\Omega)}a_{i}x_{i}\right)\right|+|f(a_{i_{0}}x_{i_{0}})|\leq\max_{i\in\Omega}|a_{i}|+\left|f\left(\sum_{i\in
A_{f}(\Omega)}a_{i}x_{i}\right)\right|$

where, as before, $A_{f}(\Omega)=\\{i\in\Omega\mid m_{t_{i}}\neq m(f)\\}$ so
that $\Omega\setminus A_{f}(\Omega)$ is at most a singleton $\\{i_{0}\\}$ with
$m_{t_{i_{0}}}=m(f)$, in which case,
$|f(x_{i_{0}})|\leq\frac{m_{t_{i_{0}}}}{w(f)}\leq 1$. Let
$\gamma=\max_{i\in\Omega}|a_{i}|$ as before and it now remains to estimate

$\displaystyle\left|f\left(\sum_{i\in A_{f}(\Omega)}a_{i}x_{i}\right)\right|$
$\displaystyle\leq\sum_{\alpha=1}^{4}\left|g_{\alpha}\left(\sum_{i\in
A_{f}(\Omega)}a_{i}x_{i}\right)\right|$
$\displaystyle\leq\frac{1}{w(f)}\left|f_{1}\left(\sum_{i\in
A_{f}(\Omega)}a_{i}x_{i}\right)\right|+\frac{1}{w(f)}\left|f_{2}\left(\sum_{i\in
A_{f}(\Omega)}a_{i}x_{i}\right)\right|$
$\displaystyle\qquad+\frac{1}{w(f)}\left|\sum_{q\in B}f_{q}\left(\sum_{i\in
A_{f}(\Omega)}a_{i}x_{i}\right)\right|+\frac{1}{w(f)}\left|\sum_{q\in
C}f_{q}\left(\sum_{i\in A_{f}(\Omega)}a_{i}x_{i}\right)\right|$

where, after applying the inductive hypothesis to the $g_{1}$ and $g_{2}$
terms and using the same argument as is used in the base case for the $g_{3}$
term, this quantity is bounded above by

$\displaystyle\frac{\gamma}{w(f)}\left(1+\frac{3}{\sqrt{w(f_{1})}}\right)+\frac{\gamma}{w(f)}\left(1+\frac{3}{\sqrt{w(f_{2})}}\right)+\frac{3\gamma}{\sqrt{w(f)}}+\frac{1}{w(f)}\left|\sum_{q\in
C}f_{q}\left(\sum_{i\in A_{f}(\Omega)}a_{i}x_{i}\right)\right|$
$\displaystyle\qquad\leq\frac{\frac{13}{5}\gamma}{w(f)}+\frac{3\gamma}{\sqrt{w(f)}}+\frac{1}{w(f)}\left|\sum_{q\in
C}f_{q}\left(\sum_{i\in A_{f}(\Omega)}a_{i}x_{i}\right)\right|$
$\displaystyle\qquad\leq\frac{(3+\frac{13}{50})\gamma}{\sqrt{w(f)}}+\frac{1}{w(f)}\left|\sum_{q\in
C}f_{q}\left(\sum_{i\in A_{f}(\Omega)}a_{i}x_{i}\right)\right|$

because $m_{1}\geq 100$. Now, note that there are sets $E_{0}\cup
E_{1}\cup\ldots\cup E_{l}$ from Lemma 4.3 such that $\\{q\mid
f_{q}\in\mathcal{C}\\}=E_{0}\cup E_{1}\cup\ldots\cup E_{l}$ and such that, for
each $1\leq z\leq l$, we have that $(f_{q})_{q\in E_{z}}$ is
$S_{n_{j_{1}}+\cdots+n_{j_{z}}}\ast\mathcal{A}_{3}$-admissible and $\max_{q\in
E_{z}}d(m(f_{q}),\lambda_{z})\leq\delta$ for some
$\lambda_{z}\in\overline{\mathcal{D}}$. Next, let $\Gamma_{z}=\\{i\in
A_{f}(\Omega)\mid m_{t_{i}}=m(f_{q})\text{ for some }q\in E_{z}\\}$ for each
$1\leq z\leq l$ and note that, if $i_{1},i_{2}\in\Gamma_{z}$, then we have
that

$\displaystyle 2^{-N}$ $\displaystyle\leq d(m_{t_{i_{1}}},m_{t_{i_{2}}})$
$\displaystyle=d(m(f_{q_{1}}),m(f_{q_{2}}))\leq
d(m(f_{q_{1}}),\lambda_{z})+d(\lambda_{z},m(f_{q_{2}}))\leq 2\delta$

for the corresponding $q_{1},q_{2}\in E_{z}$, and this cannot happen. It
follows that $\Gamma_{z}$ is at most a singleton, say $i_{z}$, for each $1\leq
z\leq l$ so that

$\displaystyle\left|\sum_{q\in\mathcal{C}}f_{q}\left(\sum_{i\in
A_{f}(\Omega)}a_{i}x_{i}\right)\right|$
$\displaystyle\leq\sum_{z=0}^{l}\sum_{q\in E_{z}}\left|f_{q}\left(\sum_{i\in
A_{f}(\Omega)\setminus\\{i_{z}\\}}a_{i}x_{i}\right)\right|+\sum_{z=0}^{l}\sum_{q\in
E_{z}}|f_{q}(a_{i_{z}}x_{i_{z}})|$ $\displaystyle\leq\sum_{z=0}^{l}\sum_{q\in
E_{z}}\frac{4\gamma}{\sqrt{w(f_{q})}}+\gamma\sum_{z=0}^{l}\sum_{q\in
E_{z}}\frac{m_{t_{i_{z}}}}{w(f_{q})}$
$\displaystyle\leq(l+1)\gamma+\sum_{z=0}^{l}\sum_{q\in
E_{z}}\frac{4\gamma}{\sqrt{w(f_{q})}}\leq(l+5)\gamma$

where again we have used the inductive hypothesis and, by putting all of these
pieces together, there is the estimate

$\displaystyle\left|f\left(\sum_{i\in\Omega}a_{i}x_{i}\right)\right|$
$\displaystyle\leq\gamma+\frac{(3+\frac{13}{50})\gamma}{\sqrt{w(f)}}+\frac{(l+5)\gamma}{w(f)}$
$\displaystyle\leq\gamma+\frac{(3+\frac{13}{50})\gamma}{\sqrt{w(f)}}+\frac{(l+5)\gamma}{10^{l}\sqrt{w(f)}}\leq\gamma+\frac{(3+\frac{13}{50}+\frac{3}{5})\gamma}{\sqrt{w(f)}}\leq\gamma+\frac{4\gamma}{\sqrt{w(f)}}$

where, in particular, $\left|f\left(\sum_{i\in
A_{f}(\Omega)}a_{i}x_{i}\right)\right|\leq\frac{4\gamma}{\sqrt{w(f)}}$ so that
the proof is complete. ∎

##  5\. Rapidly increasing sequences and basic inequality

Rapidly increasing sequences (RISs) and the basic inequality are standard
concepts used in most Banach spaces constructions using saturated norms, such
as $X_{\mathcal{D}}$. The main utility of such sequences is that they can be
found in every infinite-dimensional subspace of $X_{\mathcal{D}}$ and, via the
basic inequality, the norm of their linear combinations is bounded above by
the same linear combinations of basis vectors in the auxiliary space
$X_{\mathcal{D}}^{\mathrm{aux},\delta}$. RISs first appeared in [26] and the
basic inequality has its roots in [1]. These tools will be essential in
Section 6 when we construct, in an arbitrary infinite-dimensional block
subspace of $X_{\mathcal{D}}$, a tree of normalized vectors that witness the
failure of the Lebesgue property.

We begin with a short technical lemma and we recall that
$\text{supp}(0)=\emptyset$ and that $\emptyset<A$ and $A<\emptyset$ for every
subset $A\subset\mathbb{N}$.

######  Lemma 5.1.

Fix $\delta>0$ and assume that $f_{1}<\ldots<f_{d}\in W$ with
$2^{-\max\text{supp}(f_{1})}\leq\frac{\delta}{2}$ are
$(n_{j_{1}},\ldots,n_{j_{l}},\lambda)$-packed for some
$\lambda\in\overline{\mathcal{D}}$. Next, suppose that there exist sets of
functionals $\mathcal{F}_{1},\ldots,\mathcal{F}_{d}$ such that, for each
$1\leq q\leq d$, $\mathcal{F}_{q}=\\{g^{q}_{1},g^{q}_{2},g^{q}_{3}\\}$ with
$g^{q}_{r}\in\\{0\\}\cup W^{\text{aux},\delta}$, and:

  1. (1)

$\cup_{r=1}^{3}\text{supp}(g^{q}_{r})\subset\text{supp}(f_{q})$ and
$\text{supp}(g^{q}_{1})<\text{supp}(g^{q}_{2})<\text{supp}(g^{q}_{3})$,

  2. (2)

there is at most one $g^{q}_{\tilde{r}}\in\mathcal{F}_{q}$ such that
$g^{q}_{\tilde{r}}$ is not a unit vector,

  3. (3)

there is at most one $g^{q}_{r}\in\mathcal{F}_{q}$ whose support precedes the
support of $g^{q}_{\tilde{r}}$, and

  4. (4)

for $g^{q}_{\tilde{r}}\in\mathcal{F}_{q}$ that is not a unit vector,
$\vec{w}(g^{q}_{\tilde{r}})=\vec{w}(f_{q})$.

Then, the functionals $(g^{q}_{r})_{r=1,q=1}^{3,d}$ are
$(n_{j_{1}},\ldots,n_{j_{l}},\lambda)_{\delta}$-packed.

###### Proof.

We proceed by induction on the length $l$ of the tuple
$(m_{j_{1}},\ldots,m_{j_{l}})$. Let $l=1$ and assume that
$f_{1}<\ldots<f_{d}\in W$ with $2^{-\max\text{supp}(f_{1})}\leq\delta$ are
$(n_{j_{1}},\lambda)$-packed for some $\lambda\in\overline{\mathcal{D}}$.
Then, $(g^{q}_{r})_{r=1,q=1}^{3,d}\subset W^{\text{aux},\delta}_{k}$ for some
$k\in\mathbb{N}$ are $S_{n_{j_{1}}}\ast\mathcal{A}_{3}$-admissible and we have
that

$w(g^{q}_{\tilde{r}})=w(f_{q})\geq 2^{\max\text{supp}(f_{q-1})}\geq
2^{\max\text{supp}(\varphi)}$

for each $g^{q}_{\tilde{r}}\in\mathcal{F}_{q}$ with at least two preceding
non-zero functionals, where $\varphi$ is the non-zero functional that is two
before $g^{q}_{\tilde{r}}$. It follows that $(g^{q}_{r})_{r=1,q=1}^{3,d}$ are
skipped very fast growing in the sense of Definition 4.1. Moreover, we have
that

$d(m(g^{q}_{\tilde{r}}),\lambda)=d(m(f_{q}),\lambda)\leq
2^{-\max\text{supp}(f_{q-1})}\leq
2^{-\max\text{supp}(f_{1})}\leq\frac{\delta}{2}<\delta$

for $2\leq q\leq d$ so that $(g^{q}_{r})_{q=1,r=1}^{d,3}$ are
$(n_{j_{1}},\lambda)_{\delta}$-packed.

Assume now that the result holds for tuples of length $l$ and let
$f_{1}<\ldots<f_{d}\in W$ be
$(n_{j_{1}},\ldots,n_{j_{l}},n_{j_{l+1}},\lambda)$-packed for some
$\lambda\in\overline{\mathcal{D}}$. Let
$\mathcal{F}_{1},\ldots,\mathcal{F}_{d}$ be sets of at most three auxiliary
weighted functionals as in the statement of the lemma and let
$F_{1}<\ldots<F_{M}$ be a partition of $\\{1,\ldots,d\\}$ as in Definition
3.2. Note that $(f_{q})_{q\in F_{1}}$ are
$(n_{j_{1}},\ldots,n_{j_{l}},\lambda_{1})$-packed for some
$\lambda_{1}\in\overline{\mathcal{D}}$. It therefore follows from the
inductive hypothesis that $(g^{q}_{r})_{r=1,q\in F_{1}}^{3}$ are
$(n_{j_{1}},\ldots,n_{j_{;}},\lambda_{1})_{\delta}$-packed. Moreover, for each
$g^{q}_{\tilde{r}}$ with $q\in F_{p}$ for $p>1$, we have that

$\displaystyle d(m(g^{q}_{\tilde{r}}),\lambda)$
$\displaystyle=d(m(f_{q}),\lambda)\leq
d(m(f_{q}),\lambda_{p})+d(\lambda_{p},\lambda)$ $\displaystyle\leq 2\cdot
2^{-\max\text{supp}(f_{\max(F_{p-1})})}\leq 2\cdot
2^{-\max\text{supp}(f_{1})}\leq\delta$

and, because $(g^{q}_{r})_{q=1,r=1}^{d,3}$ are skipped very fast growing in
the sense of Definition 4.2 by the same reasoning that is used in the base
case, it follows that these functionals are
$(n_{j_{1}},\ldots,n_{j_{l}},n_{j_{l+1}},\lambda)_{\delta}$-packed so that the
proof is complete. ∎

We now recall from [5] the notion of a rapidly increasing sequence (RIS).

######  Definition 5.2.

Let $C\geq 1$, let $I\subset\mathbb{N}$ be an interval, and let $(j_{i})_{i\in
I}$ be a strictly increasing sequence of positive integers. Then, a block
sequence $(x_{i})_{i\in I}$ is said to be a $(C,(j_{i})_{i\in I})$-RIS if the
following three statements hold:

  1. (1)

$\|x_{i}\|\leq C$ for each $i\in\mathbb{N}$.

  2. (2)

$\max\text{supp}(x_{i-1})<\sqrt{m_{j_{i}}}$ for each $i\in
I\setminus\\{\min(I)\\}$.

  3. (3)

$|f(x_{i})|\leq\frac{C}{w(f)}$ for each $i\in I$ and for every $f\in W$ with
$w(f)<m_{j_{i}}$.

We recall the steps of a standard argument that shows that there exists a RIS
in every block subspace of $X_{\mathcal{D}}$. For some of the details, we
refer to specific parts of [5]. The proposition below is proved word-for-word
as [5, Proposition 6.3], and it requires only two things: Theorem 3.4 (in
place of [5, Proposition 4.1]) and that, for every $C>1$,
$m_{j}^{-1}C^{n_{j}}\to\infty$. The latter is true because, by condition (2)
on page (2),

$\displaystyle\log\left(\frac{C^{n_{j}}}{m_{j}}\right)>\frac{4\log(C)}{\log(100)}n_{j-1}\log(m_{j})-\log(m_{j})=\log(m_{j})\left(\frac{4\log(C)n_{j-1}}{\log(100)}-1\right).$

######  Proposition 5.3.

Let $Y$ be a block subspace of $X_{\mathcal{D}}$. Then, for every
$n\in\mathbb{N}$, $\varepsilon>0$, and $\delta>0$ there exists a
$(n,\varepsilon)$ special convex combination $x=\sum_{i=1}^{m}c_{i}x_{i}$ such
that $\|x\|>1/(1+\delta)$ and $x_{1},\ldots,x_{m}$ are block vectors in $Y$ of
norm at most one.

The next proposition is also proved verbatim as [5, Proposition 6.4] and it
uses Remark 4.4.

######  Proposition 5.4.

Let $j\in\mathbb{N}$, $\varepsilon>0$, and $x=\sum_{i=1}^{m}c_{i}x_{i}$ be an
$(n_{j},\varepsilon)$-special convex combination such that $\|x_{i}\|\leq 1$,
$1\leq i\leq m$. Then, for every $f\in W$ such that $w(f)<m_{j}$,

$|f(x)|\leq\frac{1+2w(f)\varepsilon}{w(f)}.$

The desired conclusion follows immediately from Proposition 5.3 and
Proposition 5.4.

######  Corollary 5.5.

For every $C>1$ and in every block subspace $Y$ of $X_{\mathcal{D}}$ there
exists a $(C,(j_{i})_{i\in\mathbb{N}})$-RIS $(x_{i})$, with $\|x_{i}\|\geq 1$,
for $i\in\mathbb{N}$.

We now prove the basic inequality that transfers the action of a weighted
functional $f\in W$ to the action of auxiliary weighted functionals on
appropriate unit vectors.

######  Proposition 5.6.

Let $(x_{i})_{i\in I}$ be a $(C,(j_{i})_{i\in I})$-RIS with
$2^{-\min\text{supp}(x_{\min(I)})}\leq\delta$. Then, for every functional
$f=\frac{1}{m_{j_{1}}\cdots m_{j_{l}}}\sum_{q=1}^{d}f_{q}\in W$, there exist
functionals $h\in\\{0\\}\cup W^{\text{aux},\delta}_{0}$ and $g\in
W^{\text{aux},\delta}$ with $\vec{w}(g)=\vec{w}(f)$ such that

$\left|f\left(\sum_{i\in I}a_{i}x_{i}\right)\right|\leq
C\left(1+\frac{1}{\sqrt{m_{j_{\min(I)}}}}\right)\left|(h+g)\left(\sum_{i\in
I}a_{i}e_{t_{i}}\right)\right|$

for every choice $(a_{i})_{i\in I}$ of scalars where
$t_{i}=\max\text{supp}(x_{i})$ for each $i\in I$.

###### Proof.

We proceed by induction on the level $k$ norming sets $W_{k}$ and the base
case is the level $k=0$ set $W_{0}=\\{\pm e_{i}^{*}\mid i\in\mathbb{N}\\}$.
Indeed, if $f=e^{*}_{i_{0}}\in W_{0}$, then

$\left|e^{*}_{i_{0}}\left(\sum_{i\in
I}a_{i}x_{i}\right)\right|\leq|a_{i_{1}}e^{*}_{i_{0}}(x_{i_{1}})|\leq
C|a_{i_{1}}|=C|e^{*}_{t_{i_{1}}}(a_{t_{i_{1}}}e_{t_{i_{1}}})|$

where $i_{0}\in\text{supp}(x_{i_{1}})$ so that the desired estimate holds with
$h=\\{0\\}$ and with $g=e^{*}_{t_{i_{1}}}$. Note that, as part of the
inductive hypothesis, we have that
$\text{supp}(h),\text{supp}(g)\subset\\{t_{i}\\}_{i\in I}$ and that: either
$h=0$ or $h\in W^{\text{aux},\delta}_{0}$,
$\text{supp}(h)<\min\text{supp}(g)$.

Assume now that the desired estimate holds for functionals in the level $k$
norming set $W_{k}$ and consider the functional

$f=\frac{1}{m_{j_{1}}\cdots m_{j_{l}}}\sum_{q=1}^{d}f_{q}\in W_{k+1}\setminus
W_{k}$

which is to say that $f_{1}<\ldots<f_{d}$ are in $W_{k}$ and are
$(n_{j_{1}},\ldots,n_{j_{l}},\lambda)$-packed for some
$\lambda\in\overline{\mathcal{D}}$. We omit if necessary the functionals
$f_{q}$ that are below the support of $x_{\min(I)}$ and it then follows that

$\frac{1}{\delta}\leq 2^{\min\text{supp}(x_{\min(I)})}\leq
2^{\max\text{supp}(f_{1})}\implies 2^{-\max\text{supp}(f_{1})}\leq\delta$

and we will use this observation later. First, define the quantity $i_{0}$ by

$i_{0}=\max\\{i\in I\mid m_{j_{1}}\cdots m_{j_{l}}\geq m_{j_{i}}\\}$

if it exists. If there does exist such an $i_{0}$, then choose
$i_{1}\in[\min(I),i_{0}]$ so that
$|a_{i_{1}}|=\max\\{|a_{i}|\\}_{i=\min(I)}^{i_{1}}$ and define the functional
$h=\text{sign}(a_{i_{1}})e^{*}_{t_{i_{1}}}\in W^{\text{aux},\delta}_{0}$. Note
that, if $\min(I)=i_{0}=i_{1}$, then

$\left|f\left(\sum_{i\leq
i_{0}}a_{i}x_{i}\right)\right|=|f(a_{i_{1}}x_{i_{1}})|\leq
C|a_{i_{1}}|=Ch(a_{i_{1}}e_{t_{i_{1}}})$

and, if $\min(I)<i_{0}$, then

$\displaystyle\left|f\left(\sum_{i\leq i_{0}}a_{i}x_{i}\right)\right|$
$\displaystyle\leq|a_{i_{0}}f(x_{i_{0}})|+\sum_{i<i_{0}}|a_{i}f(x_{i})|\leq
C|a_{i_{0}}|+\sum_{i<i_{0}}\left|\frac{a_{i}}{w(f)}\sum_{q=1}^{d}f_{q}(x_{i})\right|$
$\displaystyle\leq
C|a_{i_{1}}|+\frac{|a_{i_{1}}|}{m_{j_{i_{0}}}}\sum_{i<i_{0}}\sum_{q=1}^{d}\left|f_{q}\left(\sum_{r\in\text{supp}(x_{i})}b_{r}e_{r}\right)\right|$
$\displaystyle\leq
C|a_{i_{1}}|+\frac{|a_{i_{1}}|}{m_{j_{i_{0}}}}\sum_{i<i_{0}}\sum_{r\in\text{supp}(x_{i})}|b_{r}|\leq
C|a_{i_{1}}|+\frac{C|a_{i_{1}}|}{m_{j_{i_{0}}}}\max\text{supp}(x_{i_{0}-1})$
$\displaystyle\leq
C\left(1+\frac{1}{\sqrt{m_{j_{i_{0}-1}}}}\right)|a_{i_{1}}|\leq
C\left(1+\frac{1}{\sqrt{m_{j_{\min(I)}}}}\right)h\left(\sum_{i\leq
i_{0}}a_{i}e_{t_{i}}\right)$

and if, on the other hand, there is no such number $i_{0}$, then the sum
$\sum_{i\leq i_{0}}a_{i}x_{i}$ is empty so we take $h=0$.

Next, let $\tilde{I}=\\{i\in I\mid i>i_{0}\\}$ if such an $i_{0}$ exists and
put $\tilde{I}=I$ otherwise. We then define the subsets

$\displaystyle A$
$\displaystyle=\\{i\in\tilde{I}\mid\text{supp}(x_{i})\cap\text{supp}(f_{q})\neq\emptyset\text{
for at most one }q\\}$ $\displaystyle I_{q}$ $\displaystyle=\\{i\in
A\mid\text{supp}(x_{i})\cap\text{supp}(f_{q})\neq\emptyset\\}$ $\displaystyle
D$ $\displaystyle=\\{q\mid I_{q}\neq\emptyset\\}$ $\displaystyle B$
$\displaystyle=\tilde{I}\setminus A$

so that the intervals $I_{q}$ are pairwise disjoint and $\text{supp}(x_{i})$
intersects the support of at least two $f_{q}$ if $i\in B$. It follows by the
inductive hypothesis that, for each $q\in D$, there exist functionals
$h_{q}\in\\{0\\}\cup W^{\text{aux},\delta}_{0}$ and $g_{q}\in
W^{\text{aux},\delta}$ with $\vec{w}(g_{q})=\vec{w}(f_{q})$ so that

$\left|f_{q}\left(\sum_{i\in I_{q}}a_{i}x_{i}\right)\right|\leq
C\left(1+\frac{1}{\sqrt{m_{j_{\min(I)}}}}\right)\left|(h_{q}+g_{q})\left(\sum_{i\in
I_{q}}a_{i}e_{t_{i}}\right)\right|$

where $\text{supp}(h_{q}),\text{supp}(g_{q})\subset\text{supp}(f_{q})$ and we
have that: either $h_{q}=0$ or $h_{q}\in\\{e_{t_{i}}\\}_{i\in I_{q}}$ and
$t_{i}<\min\text{supp}(g_{q})$. Now, replace $h_{q}+g_{q}$ by
$\text{sign}\left[(h_{q}+g_{q})\left(\sum_{i\in
I_{q}}a_{i}e_{t_{i}}\right)\right](h_{q}+g_{q})$ if necessary and note that

$\displaystyle\left|f\left(\sum_{i\in\tilde{I}}a_{i}x_{i}\right)\right|$
$\displaystyle\leq\sum_{i\in B}|f(x_{i})|+\frac{1}{w(f)}\sum_{q\in
D}\left|f_{q}\left(\sum_{i\in I_{q}}a_{i}x_{i}\right)\right|$
$\displaystyle\leq\frac{C}{w(f)}\sum_{i\in
B}|a_{i}|+C\left(1+\frac{1}{\sqrt{m_{j_{\min(I)}}}}\right)\frac{1}{w(f)}\sum_{q\in
D}(h_{q}+g_{q})\left(\sum_{i\in I_{q}}a_{i}e_{t_{i}}\right)$
$\displaystyle\leq
C\left(1+\frac{1}{\sqrt{m_{j_{\min(I)}}}}\right)\frac{1}{w(f)}\left[\sum_{i\in
B}\tilde{h}_{i}+\sum_{q\in
D}(h_{q}+g_{q})\right]\left(\sum_{i\in\tilde{I}}a_{i}e_{t_{i}}\right)$

where we have used the third condition from Definition 5.2 and where
$\tilde{h}_{i}=\text{sign}(a_{i})e_{t_{i}}^{*}$ for each $i\in B$. It
therefore follows that

$\left|f\left(\sum_{i\in I}a_{i}x_{i}\right)\right|\leq
C\left(1+\frac{1}{\sqrt{m_{j_{\min(I)}}}}\right)\left[(h+g)\left(\sum_{i\in
I}a_{i}e_{t_{i}}\right)\right]$

where: either $h=0$ or $h\in\\{e_{t_{i}}\\}_{i=\min(I)}^{i_{0}}$ and
$\text{supp}(h)=t_{i_{1}}<\min\text{supp}(g)$. Indeed, we have that

$g=\frac{1}{w(f)}\left(\sum_{i\in B}\tilde{h}_{i}+\sum_{q\in
D}(h_{q}+g_{q})\right)$

so that $\vec{w}(f)=\vec{w}(g)$. It remains to prove that $g\in
W^{\text{aux},\delta}$. To that end, define for each $i\in B$ the quantity
$q_{i}=\max\\{1\leq q\leq
d\mid\min\text{supp}(f_{q})\leq\max\text{supp}(x_{i})=t_{i}\\}$ and note that
the correspondence $i\mapsto q_{i}$ is strictly increasing. If $q=q_{i}$ for
some $i\in B$, then write $\mathcal{F}_{q}=\\{h_{q},g_{q},\tilde{h}_{i}\\}$.
If $q\neq q_{i}$ for any $i\in B$, then write
$\mathcal{F}_{q}=\\{h_{q},g_{q}\\}$. Note that $\mathcal{F}_{q}$ may consist
only of $\tilde{h}_{i}$ or may be empty if $q\notin D$. It therefore follows
from Lemma 5.1 (applied to $(f_{q})_{q\in D}$) that $g\in
W^{\text{aux},\delta}$ and this completes the proof. ∎

##  6\. The failure of the Lebesgue property in subspaces of $X_{\mathcal{D}}$

In this section, we put together the results of the previous two sections in
order to prove that every subspace of $X_{\mathcal{D}}$ contains a normalized
collection $(y_{\lambda})_{\lambda\in\mathcal{D}}$ that satisfies the
estimates (1), and thus, witnesses the failure of the Lebesgue property.

######  Theorem 6.1.

Let $Y$ be an infinite-dimensional block subspace of $X_{\mathcal{D}}$. Then,
there exists a normalized collection $(y_{\lambda})_{\lambda\in\mathcal{D}}$
in $Y$ that is block with the lexicographical order of $\mathcal{D}$ such that
the following holds: for every $N\in\mathbb{N}$ and
$(\mu_{\lambda})_{\lambda\in\\{0,1\\}^{N}}$ in $\mathcal{D}$ such that, for
$\lambda\in\\{0,1\\}^{N}$, $\mu_{\lambda}\geq\lambda$ and for any scalars
$(a_{\lambda})_{\lambda\in\\{0,1\\}^{N}}$,

$\max_{\lambda\in\\{0,1\\}^{N}}|a_{\lambda}|\leq\Big{\|}\sum_{\lambda\in\\{0,1\\}^{N}}a_{\lambda}y_{\mu_{\lambda}}\Big{\|}\leq
3\max_{\lambda\in\\{0,1\\}^{N}}|a_{\lambda}|.$

In particular, the Lebesgue property fails in every infinite-dimensional
closed subspace of $X_{\mathcal{D}}$.

###### Proof.

Let $Y\subset X_{\mathcal{D}}$ and let $Z\subset Y$ be a block subspace. Then,
fix $C>1$ and let $(z_{i})$ be an infinite $(C,(j_{i})_{i\in\mathbb{N}})$-RIS
in $Z$ with $1\leq\|z_{i}\|$ for each $i\in\mathbb{N}$, which exists by
Corollary 5.5. By passing to a subsequence, we may assume that the conclusion
of Theorem 3.4 is satisfied. Recall that the nodes of $\mathcal{D}$ are in
lexicographical order with respect to $\mathcal{M}$ and choose for each weight
$m_{j}$ a number $\varepsilon_{j}\in\left(0,\frac{1}{3m_{j}2^{N_{j}}}\right)$,
where $N_{j}=|\phi(m_{j})|$.

Now, let $F_{1}<F_{2}<\ldots$ be subsets of $\mathbb{N}$ so that
$\\{\min\text{supp}(z_{i})\mid i\in F_{j}\\}$ is a maximal set in $S_{n_{j}}$
with respect to inclusion for each $j\in\mathbb{N}$. It follows from
Proposition 2.1 that there exist coefficients $(c^{j}_{i})_{i\in F_{j}}$ so
that the vector $\sum_{i\in F_{j}}c^{j}_{i}e_{i}$ is an
$(n_{j},\frac{\varepsilon_{j}}{2})$-basic scc. In particular, we have that
$x_{m_{j}}=\sum_{i\in F_{j}}c^{j}_{i}z_{i}$ is an
$(n_{j},\frac{\varepsilon_{j}}{2})$-scc and we note also that

$\|x_{m_{j}}\|=\left\|\sum_{i\in
F_{j}}c^{j}_{i}z_{i}\right\|\geq\frac{m_{1}-1}{m_{1}m_{j}}\sum_{i\in
F_{j}}c^{j}_{i}=\frac{m_{1}-1}{m_{1}m_{j}}$

from Theorem 3.4. Let $\tilde{x}_{m_{j}}=m_{j}x_{j}$ and define, for
$\lambda\in\mathcal{D}$,
$y_{\lambda}=\tilde{x}_{\phi^{-1}(m_{j})}/\|\tilde{x}_{\phi^{-1}(m_{j})}\|$ so
that $(y_{\lambda})$ is a normalized sequence in $Z$ that is block in the
lexicographical order of $\mathcal{D}$.

Now, fix $N\in\mathbb{N}$ and, for $\lambda\in\\{0,1\\}^{N}$, let
$\mu_{\lambda}\in\mathcal{D}$ such that $\mu_{\lambda}\geq\lambda$. Because
$(y_{\lambda})_{\lambda\in\mathcal{D}}$ is lexicographically block, there are
$2^{N}\leq j_{1}<\cdots<j_{2^{N}}$ such that
$\phi(\\{m_{j_{1}},\ldots,m_{j_{2^{N}}}\\})=\\{\mu_{\lambda}:\lambda\in\\{0,1\\}^{N}\\}$.
In particular, $\min\text{supp}(y_{\mu_{\lambda}})\geq N$,
$\lambda\in\\{0,1\\}^{N}$. Furthermore, for $\lambda\in\\{0,1\\}^{N}$,
$\mu_{\lambda}\geq\lambda$, and thus, for $1\leq k\neq k^{\prime}\leq 2^{N}$,
$d(m_{j_{k}},m_{j_{k^{\prime}}})\geq 2^{-N}$. Then, it follows that

$\left\|\sum_{\lambda\in\\{0,1\\}^{N}}y_{\mu_{\lambda}}\right\|=\mathscr{F}\left(\sum_{\lambda\in\\{0,1\\}^{N}}y_{\mu_{\lambda}}\right)=\mathscr{F}\left(\sum_{k=0}^{2^{N}-1}\frac{1}{\|\tilde{x}_{m_{j_{k}}}\|}m_{j_{k}}\sum_{i\in
F_{j_{k}}}c^{j_{k}}_{i}z_{i}\right)\qquad(\\#)$

for some norming functional $\mathscr{F}\in W$ and, by Proposition 5.6, this
quantity is bounded above by

$C\left|(h+g)\left(\sum_{k=0}^{2^{N}-1}\frac{1}{\|\tilde{x}_{m_{j_{k}}}\|}m_{j_{k}}\sum_{i\in
F_{j_{k}}}c^{j_{k}}_{i}e_{\rho(i)}\right)\right|\qquad(\\#\\#)$

where $h\in W^{\text{aux},\delta}_{0}\cup\\{0\\}$, $g\in
W^{\text{aux},\delta}$ with $\vec{w}(g)=\vec{w}(\mathscr{F})$, and where
$\rho(i)=\max\text{supp}(z_{i})$ for each $i\in\mathbb{N}$. Note that
$\sum_{i\in F_{j_{k}}}c^{j_{k}}_{i}e_{\rho(i)}$ is for each $0\leq k\leq
2^{N}-1$ an $(n_{j_{k}},\varepsilon_{j_{k}})$-basic scc by the observation
that immediately follows Proposition 2.1, and note also that

$\frac{m_{j_{k}}}{\|\tilde{x}_{m_{j_{k}}}\|}=\frac{1}{\|x_{m_{j_{k}}}\|}\leq\frac{m_{1}m_{j_{k}}}{m_{1}-1}.$

It therefore follows that $(\\#\\#)$ is bounded above by

$C\max_{0\leq k\leq
2^{N}-1}\frac{m_{j_{k}}\varepsilon_{j_{k}}}{\|\tilde{x}_{j_{k}}\|}+C\left|g\left(\sum_{k=0}^{2^{N}-1}\frac{1}{\|\tilde{x}_{j_{k}}\|}m_{j_{k}}\sum_{i\in
F_{j_{k}}}c^{j_{k}}_{i}e_{\rho(i)}\right)\right|\qquad(\\#\\#\\#)$

and lastly, by Proposition 4.5, we note that $(\\#\\#\\#)$ is bounded above by

$C\frac{m_{1}m_{j_{k}}}{m_{1}-1}\frac{1}{3m_{j_{k}}2^{N_{j_{k}}}}+C\left(1+\frac{4}{\sqrt{w(g)}}\right)\max_{0\leq
k\leq 2^{N}-1}\frac{1}{\|\tilde{x}_{j_{k}}\|}\leq
C\frac{100}{99}\left(1+\left(1+\frac{2}{5}\right)\right)\leq 3,$

for $C>1$ sufficiently close to one. The estimate for arbitrary scalars
coefficients follows from the 1-unconditionality of the basis of
$X_{\mathcal{D}}$. ∎

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SafeDrug: Dual Molecular Graph Encoders for Recommending Effective and Safe
Drug Combinations

TheWebConf 2021 tracking ID: 450

Part1: Cover Letter

##  1\. Main Reason for Previous Rejection

There are three reviewers (in theWebConf 2021): one reviewer suggests
acceptance, while the other two did not provide reviews due to their concern
about our manuscript’s relevance to the web conference. There are two
reviewers (in the AAAI 2021): The first reviewer encourages us to add more
processing steps, more details for datasets, and he also questions on the
experimental settings. The second reviewer thinks the motivation of the paper
is limited, and he believes that a real doctor will not consider drug molecule
structures when making prescriptions.

##  2\. New Changes

From the AAAI 2021 submission, the paper changed a lot. We improve the writing
thoroughly, so as to make it more understandable. We add more contents to the
paper, including the dataset descriptions, processing steps, metric standard
deviations, hypothesis testing results, new baseline method and additional
experiments.

For the experimental setting, we actually follow many newly published works,
such as (shang2019gamenet; zhang2017leap). The dataset we used spans several
years and includes many doctors’ practice. In fact, it is a benchmark dataset
for drug recommendation and the setting is also standard in the area. In terms
of the motivation, real doctors have no time and also are unable to use drug
molecule information for better prescription, that is why we want to develop a
computational method that can use the computer power to process useful data
sources for more accurate and effective drug recommendations. We aim to ease
the doctors’ effort while providing better clinical service for patients.

From the WebConf 2021 submission, though some of them are concerned about
fitting to CFP, the reviewers do not raise any major concern to the content
itself. We, therefore, did not change the content too much. The changes are
mostly structure-wise. Here are the changes:

First, due to space limitation, we remove the notation Table 1; we move the
Algorithm 1 to Appendix; we move the dataset processing steps, implementation
details, experimental settings, and the last two experiments to Appendix.

Then, we also made minor changes to the main text to make it more readable.
The changes can be found in all sections, including Abstract and Introduction.
For example, in subsection “Longitudinal Patient Representation”, we simplify
the original expression (During a clinical visit, doctors would first diagnose
the diseases and then apply therapeutic procedures. Based on these two steps,
the doctors are ready to prescribe drugs. In this work, the patient’s health
condition is also extracted based on both diagnosis and procedure
information.) into a simpler and clearer sentence (From longitudinal EHR data,
patient health can be encoded by their diagnosis and procedure information.)

Part2: Original Reviews (From theWebConf 2021)

##  1\. Reviewer #1

Relevance to the Web Conference:

-1: (Not relevant) This paper, despite its merits, only has tangential relevance to the Web, at best. It describes a drug recommendation algorithm. The authors ground the necessity for an ML approach to drug recommendation in the fact that ”massive biomedical data [is] available on the web.” (p.1). But this is about as far as the paper goes in establishing relevance to the conference. Given that this is also the Web and Society track of the Web Conference, I am really struggling to see how this paper could fit in.

Overall evaluation:

-2: (Reject)

Summary:

The paper provides the description of an algorithm for drug recommendation,
aimed at minimizing drug-drug interactions. The algorithm uses RNNs to
generate a both a patient representation and a drug representation the former
based on patient histories, and the latter on the drug’s molecular structure.
The authors provide experimental evidence that their algorithm improves on
what they identify as the state of the art in the field.

Originality:

3: (Creative: Only a few people in our community would have put these ideas
together) My expertise here is rather limited, but this seems like a creative
use of neural networks.

Potential impact:

2: (Limited: Impact limited to improving the state-of-the-art for the problem
being tackled) It is difficult for a non-specialist to evaluate impact of a
drug recommendation algorithm. An educated guess would lead me to believe that
drug recommendation is very difficult to do in practice, likely due to issues
that have little to do with ML per se (e.g. liability and concern about
overfitting).

Reproducibility:

2: (Some but not all code, data or the details of the experimental setup are
made available) The authors present a model that seems reasonably
straightforward to reproduce, despite its complexity.

Quality of execution:

3: (Reasonable: Generally solid work, but certain claims could be justified
better) Again, an educated guess since I am not qualified to assess
pharmacological AI, but I was able to follow the paper seen from an ML lens.

Quality of presentation:

3: (Reasonable: Understandable to a large extent, but parts of the paper need
more work) See above.

Adequacy of citations:

2: (Reasonable: Coverage of past work is acceptable, but a few papers are
missing) See above.

Ethics:

1: (No) No ethical concerns. The paper uses what appears to be a canonical
dataset in the field, and the model’s recommendations do not appear to be
currently used in medical decision making. I would have concerns about
explainability and verifiability of the model’s reasoning if these
recommendations *were* being used when making real clinical decisions, but
these are out-of-scope to this paper.

Strengths:

\+ creative use of modern neural networks methods. \+ important problem that
is likely on the list of priorities for the healthcare sector.

Weaknesses:

\- not suited to the Web Conference.

Reasons to accept:

Reasons to reject:

The biggest reason to reject is the poor fit with the conference. I had to re-
read the CfP a few times to make sure I did not misunderstand something. The
most relevant paragraph describes the requirements for health research at the
conference:

”Health on the Web: New this year, the Web and Society track also includes
research related to health on the Web (previously a separate track) that
brings together advances in Internet-related computer science with direct
benefits to the medical and health domains. Relevant research includes
advances in computer science as applied to health or medicine on the Web,
significant advances in health, utilizing existing state-of-the-art
computational techniques, and reports on significant lessons learned from a
real-world deployment of medicine or health-related Web technology. Of
particular interest is research showing how the Web can improve individual and
public or provide better healthcare delivery. Submissions related to the
Covid-19 pandemic are especially welcome.”

The submission does indeed show how advances in computer science could have
direct benefits to the medical domain. However, I do not understand which of
these advances are *Internet*-related. In my view, this paper would be better
suited to a general-purpose ML conference (e.g. AAAI), one focused on
Artificial Neural Networks (e.g. NeurIPS), or one of (what I am sure are) many
medical AI conferences.

##  2\. Reviewer #2

Relevance to the Web Conference:

1: (Relevant)

Overall evaluation:

2: (Accept)

Summary:

In this paper, the authors propose a novel SafeDrug model for the problem of
safe drug recommendations. Our model extracts rich molecule structure
information from the web and introduces dual (global and local) molecular
structural encoders to help the drug recommendation task.

Originality:

3: (Creative: Only a few people in our community would have put these ideas
together)

Potential impact:

3: (Broad: Could help ongoing research in a broader research community)

Reproducibility:

3: (All code, data, and the details of the experimental setup are made
available)

Quality of execution:

4: (Thorough: The approach is well-justified and all the claims are
convincingly supported)

Quality of presentation:

4: (Lucid: Very well written in every aspect, a pleasure to read, easy to
follow)

Adequacy of citations:

3: (Comprehensive: Can’t think of any important paper that is missed)

Ethics:

1: (No) No ethical concerns

Strengths:

  * •

SafeDrug uses a global message passing neural network (MPNN) module and a
local bipartite learning module to fully encode the connectivity and
functionality of drug molecules.

  * •

The authors claim that SafeDrug beats state of the art baselines with a
significant reduction on DDI by 19.43%, improvement over Jaccard similarity
between the recommendation and the real prescription by 2.88% and F1-score by
2.14%

  * •

SafeDrug trains faster compared to other states of the art methods, as
mentioned.

Weaknesses:

The paper lacks an in-depth conclusion with the possible improvements and its
future scope.

##  3\. Reviewer #3

Relevance to the Web Conference:

-1: (Not relevant)

Overall evaluation:

-2: (Reject)

Summary:

This work presents SafeDrug, a drug recommender system designed to provide
safe drugs combinations by taking into consideration molecular structure and
patients’ medical history.

Overall, the paper is well motivated: it mentions how existing work often take
into account previous medication, but fail to how drugs interact at a
molecular level, or are often implicitly encoded with interpretable
mechanisms. Empirical results indicate that the proposed method improves over
other techniques.

Unfortunately, this work, while of good quality, doesn’t follow into the scope
of The Web Conference. No web data or processes are modeled anywhere in the
paper; in fact, this work does not use any unstructured free form text that
could be distributed on web platforms. I would encourage the authors to submit
this work to a venue that is better suited to this topic.

Originality:

2: (Conventional: Rather straightforward, a number of people could have come
up with this)

Potential impact:

2: (Limited: Impact limited to improving the state-of-the-art for the problem
being tackled)

Reproducibility:

2: (Some but not all code, data or the details of the experimental setup are
made available)

Quality of execution:

3: (Reasonable: Generally solid work, but certain claims could be justified
better)

Quality of presentation:

3: (Reasonable: Understandable to a large extent, but parts of the paper need
more work)

Adequacy of citations:

2: (Reasonable: Coverage of past work is acceptable, but a few papers are
missing)

Ethics:

1: (No) N/A

Strengths:

Well motivated

Weaknesses:

Not relevant to the conference

Original Reviews (From AAAI 2021)

##  SafeDrug: Dual Molecular Graph Encoders for Recommending Effective and
Safe Drug Combinations

††copyright: acmcopyright††journalyear: 2021††doi:
10.1145/1122445.1122456††conference: WWW ’21: The Web Conference; April 19-23,
2021; Ljubljana, Slovenia

See pages - of figure/AAAI_sub.pdf