Most numbers are not normal

Abstract

We show, from a topological viewpoint, that most numbers are not normal in a strong sense. More precisely, the set of numbers $x \in (0,1]$ with the following property is comeager: for all integers $b\ge 2$ and $k\ge 1$ , the sequence of vectors made by the frequencies of all possibile strings of length k in the b-adic representation of x has a maximal subset of accumulation points, and each of them is the limit of a subsequence with an index set of nonzero asymptotic density. This extends and provides a streamlined proof of the main result given by Olsen (2004) in this Journal. We provide analogues in the context of analytic P-ideals and regular matrices.

1. Introduction

A real number $x \in (0,1]$ is normal if, informally, for each base $b\ge 2$ , its b-adic expansion contains every finite string with the expected uniform limit frequency (the precise definition is given in the next few lines). It is well known that most numbers x are normal from a measure theoretic viewpoint, see e.g. [ 5 ] for history and generalisations. However, it has been recently shown that certain subsets of nonnormal numbers may have full Hausdorff dimension, see e.g. [ 1, 4 ]. The aim of this work is to show that, from a topological viewpoint, most numbers are not normal in a strong sense. This provides another nonanalogue between measure and category, cf. [ 25 ].

For each $x \in (0,1]$ , denote its unique nonterminating b-adic expansion by

(1) \begin{equation}x=\sum\nolimits_{n\ge 1}\frac{d_{b,n}(x)}{b^n},\end{equation}

with each digit $d_{b,n}(x) \in \{0,1,\ldots,b-1\}$ , where $b\ge 2$ is a given integer. Then, for each string ${\boldsymbol{s}}=s_1\cdots s_k$ with digits $s_j \in \{0,1,\ldots,b-1\}$ and each $n\ge 1$ , write $\pi_{b,{\boldsymbol{s}},n}(x)$ for the proportion of strings ${\boldsymbol{s}}$ in the b-adic expansion of x which start at some position $\le n$ , i.e.,

\begin{equation*}\pi_{b,\,{\boldsymbol{s}},n}(x) \;:\!=\; \frac{\#\{i\in \{1,\ldots,n\}\;:\; d_{b,i+j-1}(x)=s_j \text{ for all }j=1,\ldots,k\}}{n}.\end{equation*}

In addition, let $S_{b}^k$ be the set of all possible strings ${\boldsymbol{s}}=s_1\cdots s_k$ in base b of length k, hence $\#S_{b}^k=b^k$ , and denote by ${\boldsymbol{\pi}}^{k}_{b,n}(x)$ the vector $(\pi_{b,{\boldsymbol{s}},n}(x)\;:\; {\boldsymbol{s}} \in S_{b}^k)$ . Of course, ${\boldsymbol{\pi}}^{k}_{b,n}(x)$ belongs to the $(b^k-1)$ -dimensional simplex for each n. However, the components of ${\boldsymbol{\pi}}^{k}_{b,n}(x)$ satisfy an additional requirement: if $k\ge 2$ and ${\boldsymbol{s}}=s_1\cdots s_{k-1}$ is a string in $S_b^{k-1}$ , then

\begin{equation*}\pi_{b,{\boldsymbol{s}},n}(x)=\sum\nolimits_{s_k}\pi_{b,{\boldsymbol{s}} s_{k},n}(x)=\sum\nolimits_{s_0}\pi_{b,s_0{\boldsymbol{s}},n}(x)+O\!\left(1/n\right) \quad \quad \text{ as }n\to \infty,\end{equation*}

where $s_0{\boldsymbol{s}}$ and ${\boldsymbol{s}}s_k$ stand for the concatened strings (indeed, the above identity is obtained by a double counting of the occurrences of the string ${\boldsymbol{s}}$ as the occurrences of all possible strings ${\boldsymbol{s}}s_k$ ; or, equivalently, as the occurrences of all possible strings $s_0{\boldsymbol{s}}$ , with the caveat of counting them correctly at the two extreme positions, hence with an error of at most 1). It follows that the set $\textrm{L}^k_{b}(x)$ of accumulation points of the sequence of vectors $({\boldsymbol{\pi}}^{k}_{b,n}(x)\;:\; n\ge 1)$ is contained in $\Delta_{b}^k$ , where

\begin{equation*}\begin{split}\Delta_{b}^k \;:\!=\; \left\{(p_{{\boldsymbol{s}}})_{{\boldsymbol{s}} \in S_{b}^k} \in \textbf{R}^{b^k}:\sum\nolimits_{{\boldsymbol{s}}} p_{{\boldsymbol{s}}}=1, \right. &p_{{\boldsymbol{s}}}\ge 0 \text{ for all }{\boldsymbol{s}} \in S_{b}^k, \\&\left.\text{ and }\sum\nolimits_{s_0}p_{s_0{\boldsymbol{s}}}=\sum\nolimits_{s_k}p_{{\boldsymbol{s}}s_k} \text{ for all }{\boldsymbol{s}} \in S_{b}^{k-1}\right\}.\end{split}\end{equation*}

Then x is said to be normal if

\begin{equation*}\forall b\ge 2, \forall k\ge 1, \forall {\boldsymbol{s}} \in S_{b}^k, \quad\lim_{n\to \infty} \pi_{b,{\boldsymbol{s}},n}(x)=1/b^{k}.\end{equation*}

Hence, if x is normal, then $\textrm{L}_{b}^k(x)=\{(1/b^{k}, \ldots, 1/b^{k})\}$ . Olsen proved in [ 23 ] that the subset of nonnormal numbers with maximal set of accumulation points is topologically large:

Theorem 1·1. The set $\{x \in (0,1]\;:\; \textrm{L}_{b}^k(x)=\Delta_{b}^k \text{ for all }b\ge 2, k\ge 1\}$ is comeager.

First, we strenghten Theorem 1·1 by showing that the set of accumulation points $\textrm{L}_{b}^k(x)$ can be replaced by the much smaller subset of accumulation points ${\boldsymbol{\eta}}$ such that every neighbourhood of ${\boldsymbol{\eta}}$ contains “sufficiently many” elements of the sequence, where “sufficiently many” is meant with respect to a suitable ideal $\mathcal{I}$ of subsets of the positive integers $\textbf{N}$ ; see Theorem 2·1. Hence, Theorem 1·1 corresponds to the case where $\mathcal{I}$ is the family of finite sets.

Then, for certain ideals $\mathcal{I}$ (including the case of the family of asymptotic density zero sets), we even strenghten the latter result by showing that each accumulation point ${\boldsymbol{\eta}}$ can be chosen to be the limit of a subsequence with “sufficiently many” indexes (as we will see in the next Section, these additional requirements are not equivalent); see Theorem 2·3. The precise definitions, together with the main results, follow in Section 2.

2. Main results

An ideal $\mathcal{I}\subseteq \mathcal{P}(\textbf{N})$ is a family closed under finite union and subsets. It is also assumed that $\mathcal{I}$ contains the family of finite sets Fin and it is different from $\mathcal{P}(\textbf{N})$ . Every subset of $\mathcal{P}(\textbf{N})$ is endowed with the relative Cantor-space topology. In particular, we may speak about $G_\delta$ -subsets of $\mathcal{P}(\textbf{N})$ , $F_\sigma$ -ideals, meager ideals, analytic ideals, etc. In addition, we say that $\mathcal{I}$ is a P-ideal if it is $\sigma$ -directed modulo finite sets, i.e., for each sequence $(S_n)$ of sets in $\mathcal{I}$ there exists $S \in \mathcal{I}$ such that $S_n\setminus S$ is finite for all $n \in \textbf{N}$ . Lastly, we denote by $\mathcal{Z}$ the ideal of asymptotic density zero sets, i.e.,

(2) \begin{equation}\mathcal{Z}=\left\{S\subseteq \textbf{N}\;:\; \textsf{d}^\star(S)=0\right\},\end{equation}

where $\textsf{d}^\star(S) \;:\!=\; \limsup_n \frac{1}{n}\#(S\cap [1,n])$ stands for the upper asymptotic density of S, see e.g. [ 20 ]. We refer to [ 14 ] for a recent survey on ideals and associated filters.

Let $x=(x_n)$ be a sequence taking values in a topological vector space X. Then we say that $\eta \in X$ is an $\mathcal{I}$ -cluster point of x if $\{n \in \textbf{N}\;:\; x_n \in U\} \notin \mathcal{I}$ for all open neighbourhoods U of $\eta$ . Note that Fin-cluster points are the ordinary accumulation points. Usually $\mathcal{Z}$ -cluster points are referred to as statistical cluster points, see e.g. [ 13 ]. It is worth noting that $\mathcal{I}$ -cluster points have been studied much before under a different name. Indeed, as it follows by [ 19 , theorem 4·2] and [ 16 , lemma 2·2], they correspond to classical “cluster points” of a filter (depending on x) on the underlying space, cf. [ 7 , definition 2, p.69].

With these premises, for each $x \in (0,1]$ and for all integers $b\ge 2$ and $k\ge 1$ , let $\Gamma_b^k (x,\mathcal{I})$ be the set of $\mathcal{I}$ -cluster points of the sequence $({\boldsymbol{\pi}}_{b,n}^k(x)\;:\;n\ge 1)$ .

Theorem 2·1. The set $\{x \in (0,1]\;:\; \Gamma_b^k (x,\mathcal{I})=\Delta_{b}^k \text{ for all }b\ge 2,k\ge 1\}$ is comeager, provided that $\mathcal{I}$ is a meager ideal.

The class of meager ideals is really broad. Indeed, it contains Fin, $\mathcal{Z}$ , the summable ideal $\{S\subseteq \textbf{N}\;:\; \sum_{n \in S}1/n<\infty\}$ , the ideal generated by the upper Banach density, the analytic P-ideals, the Fubini sum $\textrm{Fin}\times \textrm{Fin}$ , the random graph ideal, etc.; cf. e.g. [ 3, 14 ]. Note that $\Gamma_b^k (x,\mathcal{I})=\textrm{L}_b^k(x)$ if $\mathcal{I}=\textrm{Fin}$ . Therefore Theorem 2·1 significantly strenghtens Theorem 1·1.

Remark 2·2. It is not difficult to see that Theorem 2·1 does not hold without any restriction on $\mathcal{I}$ . Indeed, if $\mathcal{I}$ is a maximal ideal (i.e., the complement of a free ultrafilter on $\textbf{N})$ , then for each $x \in (0,1]$ and all integers $b\ge 2$ , $k\ge 1$ , we have that the sequence $({\boldsymbol{\pi}}_{b,n}^k(x)\;:\;n\ge 1)$ is bounded, hence it is $\mathcal{I}$ -convergent so that $\Gamma_b^k (x,\mathcal{I})$ is a singleton.

On a similar direction, if $x=(x_n)$ is a sequence taking values in a topological vector space X, then $\eta \in X$ is an $\mathcal{I}$ -limit point of x if there exists a subsequence $(x_{n_k})$ such that $\lim_k x_{n_k}=\eta$ and $\textbf{N}\setminus \{n_1,n_2,\ldots\} \in \mathcal{I}$ . Usually $\mathcal{Z}$ -limit points are referred to as statistical limit points, see e.g. [ 13 ]. Similarly, for each $x \in (0,1]$ and for all integers $b\ge 2$ and $k\ge 1$ , let $\Lambda_b^k (x,\mathcal{I})$ be the set of $\mathcal{I}$ -limit points of the sequence $({\boldsymbol{\pi}}_{b,n}^k(x)\;:\;n\ge 1)$ . The analogue of Theorem 2·1 for $\mathcal{I}$ -limit points follows.

Theorem 2·3. The set $\{x \in (0,1]\;:\; \Lambda_b^k (x,\mathcal{I})=\Delta_{b}^k \text{ for all }b\ge 2,k\ge 1\}$ is comeager, provided that $\mathcal{I}$ is an analytic P-ideal or an $F_\sigma$ -ideal.

It is known that every $\mathcal{I}$ -limit point is always an $\mathcal{I}$ -cluster point, however they can be highly different, as it is shown in [ 2 , theorem 3·1]. This implies that Theorem 2·3 provides a further improvement on Theorem 2·1 for the subfamily of analytic P-ideals.

It is remarkable that there exist $F_\sigma$ -ideals which are not P-ideals, see e.g. [ 11 , section 1·11]. Also, the family of analytic P-ideals is well understood and has been characterised with the aid of lower semicontinuous submeasures, cf. Section 3. The results in [ 6 ] suggest that the study of the interplay between the theory of analytic P-ideals and their representability may have some relevant yet unexploited potential for the study of the geometry of Banach spaces.

Finally, recalling that the ideal $\mathcal{Z}$ defined in (2) is an analytic P-ideal, an immediate consequence of Theorem 2·3 (as pointed out in the abstract) follows:

Corollary 2·4. The set of $x \in (0,1]$ such that, for all $b\ge 2$ and $k\ge 1$ , every vector in $\Delta_b^k$ is a statistical limit point of the sequence $({\boldsymbol{\pi}}_{b,n}^k(x)\;:\; n\ge 1)$ is comeager.

It would also be interesting to investigate to what extend the same results for nonnormal points belonging to self-similar fractals (as studied, e.g., by Olsen and West in [ 24 ] in the context of iterated function systems) are valid.

We leave as open question for the interested reader to check whether Theorem 2·3 can be extended for all $F_{\sigma\delta}$ -ideals including, in particular, the ideal $\mathcal{I}$ generated by the upper Banach density (which is known to not be a P-ideal, see e.g. [ 12 , p.299]).

3. Proofs of the main results

Proof of Theorem 2·1. Let $\mathcal{I}$ be a meager ideal on $\textbf{N}$ . It follows by Talagrand’s characterisation of meager ideals [ 28 , theorem 21] that it is possible to define a partition $\{I_1,I_2,\ldots\}$ of $\textbf{N}$ into nonempty finite subsets such that $S\notin \mathcal{I}$ whenever $I_n\subseteq S$ for infinitely many n. Moreover, we can assume without loss of generality that $\max I_n<\min I_{n+1}$ for all $n \in \textbf{N}$ .

The claimed set can be rewritten as $\bigcap_{b\ge 2}\bigcap_{k\ge 1}X_b^k$ , where $X_{b}^k \;:\!=\; \{x \in (0,1]\;:\; \Gamma_b^k (x,\mathcal{I})=\Delta_{b}^k\}$ . Since the family of meager subsets of (0,1] is a $\sigma$ -ideal, it is enough to show that the complement of each $X_{b}^k$ is meager. To this aim, fix $b\ge 2$ and $k\ge 1$ and denote by $\|\!\cdot\! \|$ the Euclidean norm on $\textbf{R}^{b^k}$ . Considering that $\{{\boldsymbol{\eta}}_1, {\boldsymbol{\eta}}_2, \ldots\} \;:\!=\; \Delta_b^k \cap \textbf{Q}^{b^k}$ is a countable dense subset of $\Delta_b^k$ and that $\Gamma_b^k(x,\mathcal{I})$ is a closed subset of $\Delta_b^k$ by [ 19 , lemma 3·1(iv)], it follows that

\begin{equation*}\begin{split}(0,1]\setminus X_b^k&= \bigcup\nolimits_{t\ge 1}\{x \in (0,1]\;:\; {\boldsymbol{\eta}}_t \notin \Gamma_b^k(x, \mathcal{I})\}\\&= \bigcup\nolimits_{t\ge 1}\{x \in (0,1]\;:\; \exists \varepsilon>0, \{n \in \textbf{N}\;:\; \|{\boldsymbol{\pi}}_{b,n}^k(x)-{\boldsymbol{\eta}}_t\|< \varepsilon\}\in \mathcal{I}\}\\&\subseteq \bigcup\nolimits_{t, p, m\ge 1}\{x \in (0,1]\;:\; \forall q\ge p, \exists n \in I_q, \|{\boldsymbol{\pi}}_{b,n}^k(x)-{\boldsymbol{\eta}}_t\|\ge\; {1}/{m} \}.\\\end{split}\end{equation*}

Denote by $S_{t,m,p}$ the set in the latter union. Thus it is sufficient to show that each $S_{t,p,m}$ is nowhere dense. To this aim, fix $t,p,m \in \textbf{N}$ and a nonempty relatively open set $G\subseteq (0,1]$ . We claim there exists a nonempty open set U contained in G and disjoint from $S_{t,p,m}$ . Since G is nonempty and open in (0, 1], there exists a string $\tilde{{\boldsymbol{s}}}=s_1\cdots s_j \in S_b^j$ such that $x \in G$ whenever $d_{b,i}(x)=s_i$ for all $i=1,\ldots,j$ . Now, pick $x^\star \in (0,1]$ such that $\lim_n {\boldsymbol{\pi}}_{b,n}^k(x^\star)={\boldsymbol{\eta}}_t$ , which exists by [ 22 , theorem 1]. In addition, we can assume without loss of generality that $d_{b,i}(x^\star)=s_i$ for all $i=1,\ldots,j$ . Since ${\boldsymbol{\pi}}_{b,n}^k(x^\star)$ is convergent to ${\boldsymbol{\eta}}_t$ , there exists $q \ge p+j$ such that $\|{\boldsymbol{\pi}}^k_{b,n}(x^\star)-{\boldsymbol{\eta}}_{t} \|< \;{1}/{m}$ for all $n \ge \min I_{q}$ . Define $V \;:\!=\; \{x \in (0,1]\;:\; d_{b,i}(x)=d_{b,i}(x^\star) \text{ for all }i=1,\ldots,\max I_{q}+k\}$ and note that $V\subseteq G$ because $d_{b,i}(x)=s_i$ for all $i\le j$ and $x \in V$ , and $V\cap S_{t,m,p}=\emptyset$ because, for each $x\in V$ , the required property is not satisfied for this choice of q since ${\boldsymbol{\pi}}_{b,n}^k(x)={\boldsymbol{\pi}}_{b,n}^k(x^\star)$ for all $n \le \max I_q$ . Clearly, V has nonempty interior, hence it is possible to choose such $U\subseteq V$ .

This proves that each $S_{t,m,p}$ is nowhere dense, concluding the proof.

Before we proceed to the proof of Theorem 2·3, we need to recall the classical Solecki’s characterisation of analytic P-ideals. A lower semicontinuous submeasure (in short, lscsm) is a monotone subadditive function $\varphi\;:\; \mathcal{P}(\textbf{N}) \to [0,\infty]$ such that $\varphi(\emptyset)=0$ , $\varphi(\{n\})<\infty$ , and $\varphi(A)=\lim_m \varphi(A\cap [1,m])$ for all $A\subseteq \textbf{N}$ and $n \in \textbf{N}$ . It follows by [ 26 , theorem 3·1] that an ideal $\mathcal{I}$ is an analytic P-ideal if and only if there exists a lscsm $\varphi$ such that

(3) \begin{equation}\mathcal{I}=\{A\subseteq \textbf{N}\;:\; \|A\|_\varphi=0\},\,\,\, \|\textbf{N}\|_\varphi =1,\,\,\,\text{ and }\,\,\varphi(\textbf{N})<\infty.\end{equation}

Here, $\|A\|_\varphi \;:\!=\; \lim_n \varphi(A\setminus [1,n])$ for all $A\subseteq \textbf{N}$ . Note that $\|A\|_\varphi=\|B\|_\varphi$ whenever the symmetric difference $A\bigtriangleup B$ is finite, cf. [ 11 , lemma 1·3·3(b)]. Easy examples of lscsms are $\varphi(A) \;:\!=\; \# A$ or $\varphi(A) \;:\!=\; \sup_n \!({1}/{n})\#(A \cap [1,n])$ for all $A\subseteq \textbf{N}$ which lead, respectively, to the ideals Fin and $\mathcal{Z}$ through the representation (3).

Proof of Theorem 2·3. First, let us suppose that $\mathcal{I}$ is an $F_\sigma$ -ideal. We obtain by [ 2 , theorem 2·3] that $\Lambda_b^k (x,\mathcal{I})=\Gamma_b^k (x,\mathcal{I})$ for each $b\ge 2$ , $k\ge 1$ , and $x \in (0,1]$ . Therefore the claim follows by Theorem 2·1.

Then, we assume hereafter that $\mathcal{I}$ is an analytic P-ideal generated by a lscsm $\varphi$ as in (3). Fix integers $b\ge 2$ and $k\ge 1$ , and define the function

\begin{equation*}\mathfrak{u}\;:\; (0,1]\times \Delta_b^k \longrightarrow \textbf{R}\;:\; (x,{\boldsymbol{\eta}}) \longmapsto \lim_{t\to \infty} \|\{n \in \textbf{N}\;:\; \|{\boldsymbol{\pi}}_{b,n}^k(x)-{\boldsymbol{\eta}}\|\le \; {1}/{t} \}\|_\varphi{,} \end{equation*}

where $\|\!\cdot\!\|$ stands for the Euclidean norm on $\textbf{R}^{b^k}$ . It follows by [ 2 , lemma 2·1] that every section $\mathfrak{u}(x,\cdot)$ is upper semicontinuous, so that the set

\begin{equation*}\Lambda_b^k(x,\mathcal{I},q) \;:\!=\; \{{\boldsymbol{\eta}} \in \Delta_b^k \;:\; \mathfrak{u}(x,{\boldsymbol{\eta}}) \ge q\}\end{equation*}

is closed for each $x \in (0,1]$ and $q \in \textbf{R}$ .

At this point, we prove that, for each ${\boldsymbol{\eta}} \in \Delta_b^k$ , the set $X({\boldsymbol{\eta}}) \;:\!=\; \{x \in (0,1]\;:\; \mathfrak{u}(x,{\boldsymbol{\eta}}) \ge\; {1}/{2}\}$ is comeager. To this aim, fix ${\boldsymbol{\eta}} \in \Delta_b^k$ and notice that

\begin{equation*}\begin{split}(0,1]\setminus X({\boldsymbol{\eta}})&=\bigcup\nolimits_{t\ge 1}\{x \in (0,1]\;:\; \|\{n \in \textbf{N}\;:\; \|{\boldsymbol{\pi}}_{b,n}^k(x)-{\boldsymbol{\eta}}\|\le\; {1}/{t}\}\|_\varphi < \; {1}/{2}\}\\&=\bigcup\nolimits_{t\ge 1}\{x \in (0,1]\;:\; \lim_{h\to \infty} \varphi(\{n\ge h\;:\; \|{\boldsymbol{\pi}}_{b,n}^k(x)-{\boldsymbol{\eta}}\|\le\; {1}/{t}\})< \; {1}/{2}\}\\&=\bigcup\nolimits_{t,h\ge 1}\{x \in (0,1]\;:\; \varphi(\{n\ge h \;:\; \|{\boldsymbol{\pi}}_{b,n}^k(x)-{\boldsymbol{\eta}}\|\le \; {1}/{t}\})< \; {1}/{2}\}.\end{split}\end{equation*}

Denoting by $Y_{t,h}$ the inner set above, it is sufficient to show that each $Y_{t,h}$ is nowhere dense. Hence, fix $G\subseteq (0,1]$ , $\tilde{{\boldsymbol{s}}} \in S_b^j$ , and $x^\star \in (0,1]$ as in the proof of Theorem 2·1. Considering that $\|\!\cdot\!\|_\varphi$ is invariant under finite sets, it follows that

\begin{equation*}\varphi(\{n\ge j^{\prime} \;:\; \|{\boldsymbol{\pi}}_{b,n}^k(x^\star)-{\boldsymbol{\eta}}\|\le \; {1}/{t}\})\ge \|\{n\ge j^{\prime} \;:\; \|{\boldsymbol{\pi}}_{b,n}^k(x^\star)-{\boldsymbol{\eta}}\|\le \; {1}/{t}\}\|_\varphi=\mathfrak{u}(x^\star, {\boldsymbol{\eta}})=1,\end{equation*}

where $j^\prime \;:\!=\; j+h$ . Since $\varphi$ is lower semicontinuous, there exists an integer $j^{\prime\prime}>j^\prime$ such that

\begin{equation*}\varphi(\{n\in [j^\prime, j^{\prime\prime}]\;:\; \|{\boldsymbol{\pi}}_{b,n}^k(x^\star)-{\boldsymbol{\eta}}\|\le \; {1}/{t}\})\ge\; {1}/{2}.\end{equation*}

Define $V \;:\!=\; \{x \in (0,1]\;:\; d_{b,i}(x)=d_{b,i}(x^\star) \text{ for all }i=1,\ldots,j^{\prime\prime}\}$ . Similarly, note that $V\subseteq G$ because $d_{b,i}(x)=s_i$ for all $i\le j$ and $x \in V$ , and $V \cap Y_{t,h}=\emptyset$ because $\varphi(\{n\ge h\;:\; \|{\boldsymbol{\pi}}_{b,n}^k(x)-{\boldsymbol{\eta}}\|\le\; {1}/{t}\})$ is at least $\varphi(\{n\in [j^\prime, j^{\prime\prime}]\;:\; \|{\boldsymbol{\pi}}_{b,n}^k(x)-{\boldsymbol{\eta}}\|\le\; {1}/{t}\})\ge \; {1}/{2}$ for all $x \in V$ . Since V has nonempty interior, it is possible to choose $U\subseteq V$ with the required property.

Finally, let E be a countable dense subset of $\Delta_b^k$ . Considering that $X \;:\!=\; \{x \in (0,1]\;:\; E\subseteq \Lambda_b^k(x,\mathcal{I},\; {1}/{2})\}$ is equal to $\bigcap_{{\boldsymbol{\eta}} \in E}X({\boldsymbol{\eta}})$ , it follows that the set $X$ is comeager. However, considering that

\begin{equation*}\Lambda_b^k(x,\mathcal{I})=\bigcup\nolimits_{q>0}\Lambda_b^k(x,\mathcal{I},q)\end{equation*}

by [ 2 , theorem 2·2] and that $\Lambda_b^k(x,\mathcal{I},\; {1}/{2})$ is a closed subset such that $E\subseteq \Lambda_b^k(x,\mathcal{I},\; {1}/{2})\subseteq \Lambda_b^k(x,\mathcal{I})\subseteq \Delta_b^k$ for all $x \in X$ , we obtain that $\Lambda_b^k(x,\mathcal{I}, \; {1}/{2})=\Lambda_b^k(x,\mathcal{I})=\Delta_b^k$ for all $x \in X$ . In particular, the claimed set contains X, which is comeager. This concludes the proof.

4. Applications

4·1. Hausdorff and packing dimensions

We refer to [ 10 , chapter 3] for the definitions of the Hausdorff dimension and the packing dimension.

Proposition 4·1. The sets defined in Theorem 2·1 and Theorem 2·3 have Hausdorff dimension 0 and packing dimension 1.

Proof. Reasoning as in [ 23 ], the claimed sets are contained in the corresponding ones with ideal Fin, which have Hausdorff dimension 0 by [ 22 , theorem 2·1]. In addition, since all sets are comeager, we conclude that they have packing dimension 1 by [ 10 , corollary 3·10(b)].

4·2. Regular matrices

We extend the main results contained in [ 15, 27 ]. To this aim, let $A=(a_{n,i}\;:\; n,i \in \textbf{N})$ be a regular matrix, that is, an infinite real-valued matrix such that, if ${\boldsymbol{z}}=({\boldsymbol{z}}_n)$ is a $\textbf{R}^d$ -valued sequence convergent to ${\boldsymbol{\eta}}$ , then $A_n{\boldsymbol{z}} \;:\!=\; \sum_i a_{n,i}{\boldsymbol{z}}_i$ exists for all $n \in \textbf{N}$ and $\lim_n A_n{\boldsymbol{z}}={\boldsymbol{\eta}}$ , see e.g. [ 9 , chapter 4]. Then, for each $x \in (0,1]$ and integers $b\ge 2$ and $k\ge 1$ , let $\Gamma_{b}^k(x,\mathcal{I},A)$ be the set of $\mathcal{I}$ -cluster points of the sequence of vectors $\left(A_n{\boldsymbol{\pi}}_{b}^k(x)\;:\; n \ge 1\right)$ , where ${\boldsymbol{\pi}}_{b}^k(x)$ is the sequence $({\boldsymbol{\pi}}_{b,n}^k(x)\;:\; n\ge 1)$ .

In particular, $\Gamma_{b}^k(x,\mathcal{I},A)=\Gamma_{b}^k(x,\mathcal{I})$ if A is the infinite identity matrix.

Theorem 4·2. The set $\{x \in (0,1]\;:\; \Gamma_b^k (x,\mathcal{I},A)\supseteq \Delta_{b}^k \text{ for all }b\ge 2, k\ge 1\}$ is comeager, provided that $\mathcal{I}$ is a meager ideal and A is a regular matrix.

Proof. Fix a regular matrix $A=(a_{n,i})$ and a meager ideal $\mathcal{I}$ . The proof goes along the same lines as the proof of Theorem 2·1, replacing the definition of $S_{t,m,p}$ with

\begin{equation*}S^\prime_{t,m,p} \;:\!=\; \{x \in (0,1]\;:\;\forall q\ge p, \exists n \in I_q,\|A_n{\boldsymbol{\pi}}^k_{b}(x)-{\boldsymbol{\eta}}_{t} \|\ge \; {1}/{m} \}.\end{equation*}

Recall that, thanks to the classical Silverman–Toeplitz characterisation of regular matrices, see e.g. [ 9 , theorem 4·1, II] or [ 8 ], we have that $\sup_n \sum_i |a_{n,i}|<\infty$ . Since $\lim_n {\boldsymbol{\pi}}_{b,n}^k(x^\star)={\boldsymbol{\eta}}_t$ , it follows that there exist sufficiently large integers $q\ge p+j$ and $j_A \ge j$ such that, if $d_{b,i}(x)=d_{b,i}(x^\star)$ for all $i=1,\ldots,j_A+k$ , then

(4) \begin{equation}\begin{split}\|A_n{\boldsymbol{\pi}}^k_{b}(x)-{\boldsymbol{\eta}}_{t} \|&\le \|A_n{\boldsymbol{\pi}}^k_{b, }(x^\star)-{\boldsymbol{\eta}}_{t}\|+\left\|\sum\nolimits_i a_{n,i}({\boldsymbol{\pi}}^k_{b,i}(x)-{\boldsymbol{\pi}}^k_{b,i}(x^\star))\right\|\\&\le \|A_n{\boldsymbol{\pi}}^k_{b }(x^\star)-{\boldsymbol{\eta}}_{t} \|+\sum\nolimits_i |a_{n,i}| \, \|{\boldsymbol{\pi}}^k_{b,i}(x)-{\boldsymbol{\pi}}^k_{b,i}(x^\star)\|\\&\le \|A_n{\boldsymbol{\pi}}^k_{b}(x^\star)-{\boldsymbol{\eta}}_{t}\|+\sum\nolimits_{i> j_A} |a_{n,i}|< \frac{1}{m}\end{split}\end{equation}

for all $n \in I_{q}$ . We conclude analogously that $S^\prime_{t,m,p}$ is nowhere dense.

The main result in [ 27 ] corresponds to the case $\mathcal{I}=\textrm{Fin}$ and $k=1$ , although with a different proof; cf. also Example 4·10 below.

At this point, we need an intermediate result which is of independent interest. For each bounded sequence ${\boldsymbol{x}}=({\boldsymbol{x}}_n)$ with values in $\textbf{R}^k$ , let $\textrm{K}\text{-}\textrm{core}({\boldsymbol{x}})$ be the Knopp core of ${\boldsymbol{x}}$ , that is, the convex hull of the set of accumulation points of ${\boldsymbol{x}}$ . In other words, $\textrm{K}\text{-}\textrm{core}({\boldsymbol{x}})=\textrm{co}\, \textrm{L}_{{\boldsymbol{x}}}$ , where $\textrm{co}\, S$ is the convex hull of $S\subseteq \textbf{R}^k$ and $\textrm{L}_{{\boldsymbol{x}}}$ is the set of accumulation points of ${\boldsymbol{x}}$ . The ideal version of the Knopp core has been studied in [ 16, 18 ]. The classical Knopp theorem states that, if $k=2$ and A is a nonnegative regular matrix, then

(5) \begin{equation}\textrm{K}\text{-}\textrm{core}(A{\boldsymbol{x}})\subseteq \textrm{K}\text{-}\textrm{core}({\boldsymbol{x}})\end{equation}

for all bounded sequences ${\boldsymbol{x}}$ , where $A{\boldsymbol{x}}=(A_n{\boldsymbol{x}}\;:\; n\ge 1)$ , see [ 17 , p. 115]; cf. [ 9 , chapter 6] for a textbook exposition. A generalisation in the case $k=1$ can be found in [ 21 ]. We show, in particular, that a stronger version of Knopp’s theorem holds for every $k \in \textbf{N}$ .

Proposition 4·3. Let ${\boldsymbol{x}}=({\boldsymbol{x}}_n)$ be a bounded sequence taking values in $\mathbf{R}^k$ , and fix a regular matrix A such that $\lim_n \sum_i |a_{n,i}|=1$ . Then inclusion (5) holds.

Proof. Define $\kappa \;:\!=\; \sup_n \|{\boldsymbol{x}}_n\|$ and let ${\boldsymbol{\eta}}$ be an accumulation point of $A{\boldsymbol{x}}$ . It is sufficient to show that ${\boldsymbol{\eta}}\in K \;:\!=\; \textrm{K}\text{-}\textrm{core}({\boldsymbol{x}})$ . Possibly deleting some rows of A, we can assume without loss of generality that $\lim A{\boldsymbol{x}}={\boldsymbol{\eta}}$ . For each $m \in \textbf{N}$ , let $K_m$ be the closure of $\textrm{co}\{x_m,x_{m+1},\ldots\}$ , hence $K\subseteq K_m$ . Define $d({\boldsymbol{a}}, C) \;:\!=\; \min_{{\boldsymbol{b}} \in C} \|{\boldsymbol{a}}-{\boldsymbol{b}}\|$ for all ${\boldsymbol{a}} \in \textbf{R}^k$ and nonempty compact sets $C\subseteq \textbf{R}^k$ . In addition, for each $m \in \textbf{N}$ , let $Q_m({\boldsymbol{a}})\in K_m$ be the unique vector such that $d({\boldsymbol{a}}, K_m)=\|{\boldsymbol{a}}-Q_m({\boldsymbol{a}})\|$ . Similarly, let $Q({\boldsymbol{a}})$ be the vector in K which minimizes its distance with ${\boldsymbol{a}}$ . Then, notice that, for all $n,m \in \textbf{N}$ , we have

\begin{equation*}\begin{split}d(A_n{\boldsymbol{x}}, K) &\le \inf\nolimits_{{\boldsymbol{b}} \in K} \inf\nolimits_{{\boldsymbol{c}} \in \textbf{R}^k} (\|A_n{\boldsymbol{x}}-{\boldsymbol{c}}\|+\|{\boldsymbol{c}}-{\boldsymbol{b}}\|) \\&\le \inf\nolimits_{{\boldsymbol{c}} \in K_m} \inf\nolimits_{{\boldsymbol{b}} \in K} (\|A_n{\boldsymbol{x}}-{\boldsymbol{c}}\|+\|{\boldsymbol{c}}-{\boldsymbol{b}}\|) \\&\le \inf\nolimits_{{\boldsymbol{c}} \in K_m} \|A_n{\boldsymbol{x}}-{\boldsymbol{c}}\|+\sup\nolimits_{{\boldsymbol{y}}\in K_m} \inf\nolimits_{{\boldsymbol{b}} \in K}\|{\boldsymbol{y}}-{\boldsymbol{b}}\| \\&= d(A_n{\boldsymbol{x}},K_m)+\sup\nolimits_{{\boldsymbol{y}}\in K_m}d({\boldsymbol{y}},K).\end{split}\end{equation*}

Since $d({\boldsymbol{\eta}},K)=\lim_n d(A_n{\boldsymbol{x}}, K)$ by the continuity of $d(\cdot,K)$ , it is sufficient to show that both $d(A_n{\boldsymbol{x}}, K_m)$ and $\sup_{{\boldsymbol{y}} \in K_m} d({\boldsymbol{y}},K)$ are sufficiently small if n is sufficiently large and m is chosen properly.

To this aim, fix $\varepsilon >0$ and choose $m \in \textbf{N}$ such that $\sup\nolimits_{{\boldsymbol{y}}\in K_m}d({\boldsymbol{y}},K)\le\; {\varepsilon}/{2}$ . Indeed, it is sufficient to choose $m \in \textbf{N}$ such that $d({\boldsymbol{x}}_n, \textrm{L}_{{\boldsymbol{x}}})< \; {\varepsilon}/{2}$ for all $n\ge m$ : indeed, in the opposite, the subsequence $({\boldsymbol{x}}_j)_{j \in J}$ , where $J \;:\!=\; \{n\in \textbf{N}\;:\; d({\boldsymbol{x}}_n, \textrm{L}_{{\boldsymbol{x}}})\ge \; {\varepsilon}/{2}\}$ , would be bounded and without any accumulation point, which is impossible. Now pick ${\boldsymbol{y}} \in K_m$ so that ${\boldsymbol{y}}=\sum_j \lambda_{i_j} {\boldsymbol{x}}_{i_j}$ for some strictly increasing sequence $(i_j)$ of positive integers such that $i_1 \ge m$ and some real nonnegative sequence $(\lambda_{i_j})$ with $\sum_{j} \lambda_{i_j}=1$ . It follows that

\begin{equation*}d({\boldsymbol{y}},K)\le \left\|{\boldsymbol{y}}-\sum\nolimits_{j}\lambda_{i_j}Q({\boldsymbol{x}}_{i_j}) \right\|\le \sum\nolimits_{j}\lambda_{i_j} \left\|{\boldsymbol{x}}_{i_j}-Q({\boldsymbol{x}}_{i_j})\right\|\le \sum\nolimits_{j}\lambda_{i_j} d({\boldsymbol{x}}_{i_j}, L_{{\boldsymbol{x}}})\le \frac{\varepsilon}{2}.\end{equation*}

Suppose for the moment that A has nonnegative entries. Since A is regular, we get $\lim_n \sum_i a_{n,i}=1$ and $\lim_n \sum_{i<m}a_{n,i}=0$ by the Silverman–Toeplitz characterisation, hence $\lim_n\sum_{i\ge m} a_{n,i}=1$ and there exists $n_0 \in \textbf{N}$ such that $\sum_{i\ge m} a_{n,i} \ge \; {1}/{2}$ for all $n\ge n_0$ . Thus, for each $n\ge n_0$ , we obtain that $ d(A_n{\boldsymbol{x}},K_m)=\|A_n{\boldsymbol{x}}-Q_m(A_n{\boldsymbol{x}})\| \le \alpha_n+\beta_n+\gamma_n,$ where

\begin{equation*}\alpha_n \;:\!=\; \left\| A_n{\boldsymbol{x}}-\frac{A_n{\boldsymbol{x}}}{\sum_i a_{n,i}}\right\|, \quad \beta_n \;:\!=\; \left\| \frac{A_n{\boldsymbol{x}}}{\sum_i a_{n,i}}-Q_m\!\left(\frac{A_n{\boldsymbol{x}}}{\sum_i a_{n,i}}\right)\right\|\end{equation*}

and

\begin{equation*}\gamma_n \;:\!=\; \left\| Q_m\left(\frac{A_n{\boldsymbol{x}}}{\sum_i a_{n,i}}\right)-Q_m(A_n{\boldsymbol{x}})\right\|.\end{equation*}

Recalling that $\kappa=\sup_n \|{\boldsymbol{x}}_n\|$ , it is easy to see that

\begin{equation*}\gamma_n \le \alpha_n\le \kappa \sum\nolimits_i |a_{n,i}| \cdot \left(1-\frac{1}{\sum\nolimits_i a_{n,i}}\right).\end{equation*}

In addition, setting $t_n \;:\!=\; \sum_{i\ge m}a_{n,i}/\sum_i a_{n,i} \in [0,1]$ for all $n\ge n_0$ , we get

(6) \begin{equation}\begin{split}\beta_n&\le \left\| \frac{\sum_i^\star}{\sum_i a_{n,i}}-\frac{\sum_{i\ge m}^\star}{\sum_{i\ge m} a_{n,i}}\right\|\\&=\frac{1}{\sum\nolimits_{i\ge m}a_{n,i}\sum\nolimits_{i}a_{n,i}}\left\| \sum\nolimits_{i\ge m}a_{n,i}\left(\sum\nolimits_{i< m}^\star+\sum\nolimits_{i\ge m}^\star \right)-\sum\nolimits_ia_{n,i}\sum\nolimits_{i\ge m}^\star\right\|\\&=\frac{1}{\sum\nolimits_{i\ge m}a_{n,i}}\left\| t_n\sum\nolimits_{i< m}^\star +(1-t_n)\sum\nolimits_{i\ge m}^\star\right\|\\&\le 2\kappa\!\left( t_n \sum\nolimits_{i<m} |a_{n,i}|+(1-t_n)\sum\nolimits_{i} |a_{n,i}|\right),\end{split}\end{equation}

where $\sum_{i \in I}^\star$ stands for $\sum_{i \in I}a_{n,i}{\boldsymbol{x}}_i$ . Note that the hypothesis that the entries of A are nonnegative has been used only in the first line of (6), so that $\sum^\star_{i\ge m}/\sum_{i\ge m}a_{n,i} \in K_m$ . Since $\lim_n \sum_{i<m}|a_{n,i}|=0$ , $\lim_n t_n=1$ , and $\sup_n \sum_i |a_{n,i}|<\infty$ by the regularity of A, it follows that all $\alpha_n, \beta_n, \gamma_n$ are smaller than ${\varepsilon}/{6}$ if n is sufficiently large. Therefore $d(A_n{\boldsymbol{x}},K)\le \varepsilon$ and, since $\varepsilon$ is arbitrary, we conclude that ${\boldsymbol{\eta}}=\lim_n A_n{\boldsymbol{x}} \in K$ .

Lastly, suppose that A is a regular matrix such that $\lim_n \sum_i |a_{n,i}|=1$ and let $B=(b_{n,i})$ be the nonnegative regular matrix defined by $b_{n,i}=|a_{n,i}|$ for all $n,i \in \textbf{N}$ . Considering that

\begin{equation*}d(A_n{\boldsymbol{x}}, K_m) \le \|A_n{\boldsymbol{x}}- B_n {\boldsymbol{x}}\|+d(B_n{\boldsymbol{x}}, K_m) \le \kappa \sum\nolimits_i |a_{n,i}-|a_{n,i}||+\varepsilon,\end{equation*}

and that $\lim_n\sum\nolimits_i |a_{n,i}-|a_{n,i}||= 0$ because $\lim_n \sum_i a_{n,i}=\lim_n\sum_i |a_{n,i}|=1$ , we conclude that $d(A_n{\boldsymbol{x}}, K_m) \le 2\varepsilon$ whenever n is sufficiently large. The claim follows as before.

The following corollary is immediate:

Corollary 4·4. Let ${\boldsymbol{x}}=({\boldsymbol{x}}_n)$ be a bounded sequence taking values in $\mathbf{R}^k$ , and fix a nonnegative regular matrix A. Then inclusion (5) holds.

Remark 4·5. Inclusion (5) fails for an arbitrary regular matrix: indeed, let $A=(a_{n,i})$ be the matrix defined by $a_{n,2n}=2$ , $a_{n,2n-1}=-1$ for all $n \in \textbf{N}$ , and $a_{n,i}=0$ otherwise. Set also $k=1$ and let x be the sequence such that $x_n=(\!-\!1)^n$ for all $n\in \textbf{N}$ . Then A is regular and $\lim Ax=3 \notin \{-1,1\}=\textrm{K}\text{-}\textrm{core}(x)$ .

Remark 4·6. Proposition 4·3 keeps holding on a (possibly infinite dimensional) Hilbert space X with the following provisoes: replace the definition of $\textrm{K}\text{-}\textrm{core}({\boldsymbol{x}})$ with the closure of $\textrm{co}\,\textrm{L}_{{\boldsymbol{x}}}$ (this coincides in the case that $X=\textbf{R}^k$ ) and assume that the sequence ${\boldsymbol{x}}$ is contained in a compact set (so that $\textrm{K}\text{-}\textrm{core}({\boldsymbol{x}})$ is also nonempty).

With these premises, we can strenghten Theorem 4·2 as follows.

Theorem 4·7. The set $\{x \in (0,1]\;:\; \Gamma_b^k (x,\mathcal{I},A)=\Delta_{b}^k \text{ for all }b\ge 2, k\ge 1\}$ is comeager, provided that $\mathcal{I}$ is a meager ideal and A is a regular matrix such that $\lim_n\sum_i |a_{n,i}|=1$ .

Proof. Let us suppose that $A=(a_{n,i})$ is nonnegative regular matrix, i.e., $a_{n,i}\ge 0$ for all $n,i \in \textbf{N}$ , and fix a meager ideal $\mathcal{I}$ , a real $x \in (0,1]$ , and integers $b\ge 2$ , $k\ge 1$ . Thanks to Theorem 4·2, it is sufficient to show that every accumulation point of the sequence $(A_n{\boldsymbol{\pi}}_{b}^k(x)\;:\; n\ge 1)$ is contained in the convex hull of the set of accumulation points of $({\boldsymbol{\pi}}_{b,n}^k(x)\;:\; n\ge 1)$ , which is in turn contained into $\Delta_b^k$ . This follows by Proposition 4·3.

Since the family of meager sets is a $\sigma$ -ideal, the following is immediate by Theorem 4·7.

Corollary 4·8. Let $\mathscr{A}$ be a countable family of regular matrices such that $\lim_n\sum_i |a_{n,i}|=1$ . Then the set $\{x \in (0,1]\;:\; \Gamma_b^k (x,\mathcal{I},A)=\Delta_{b}^k \text{ for all }b\ge 2, k\ge 1, \text{ and all }A \in \mathscr{A}\,\}$ is comeager, provided that $\mathcal{I}$ is a meager ideal.

It is worth to remark that the main result [ 15 ] is obtained as an instance of Corollary 4·8, letting $\mathscr{A}$ be the set of iterates of the Cesàro matrix (note that they are nonnegative regular matrices), and setting $k=1$ and $\mathcal{I}=\textrm{Fin}$ . The same holds for the iterates of the Hölder matrix and the logarithmic Riesz matrix as in [ 24 , sections 3 and 4].

Next, we show that the hypothesis $\lim_n \sum_i |a_{n,i}|=1$ for the entries of the regular matrix in Theorem 4·7 cannot be removed.

Example 4·9. Let $A=(a_{n,i})$ be the matrix such that $a_{n,(2n-1)!}=-1$ and $a_{n,(2n)!}=2$ for all $n \in \textbf{N}$ , and $a_{n,i}=0$ otherwise. It is easily seen that A is regular. Then, set $b=2$ , $k=1$ , and $\mathcal{I}=\textrm{Fin}$ . We claim that the set of all $x \in (0,1]$ such that 2 is an accumulation point of the sequence $\pi_{2,1}(x)=(\pi_{2,1,n}(x)\;:\; n\ge 1)$ is comeager. Indeed, its complement can be rewritten as $\bigcup_{m,p}S_{m,p}$ , where

\begin{equation*}S_{m,p} \;:\!=\; \{x \in (0,1]\;:\; |A_n\pi_{2,1}(x)-2|\ge \; {1}/{m} \text{ for all }n\ge p\}.\end{equation*}

Let $x^\star\in (0,1]$ such that $d_{2,n}(x^\star)=1$ if and only if $(2i-1)!\le n<(2i)!$ for some $i \in \textbf{N}$ . Then it is easily seen that $\lim_n \pi_{2,1,n}(x^\star)=2$ . Along the same lines of the proof of Theorem 4·2, it follows that each $S_{m,p}$ is meager. We conclude that $\{x \in (0,1]\;:\; \Gamma_2^1 (x,\textrm{Fin},A)=\Delta_{2}^1\}$ is meager, which proves that the condition $\lim_n \sum_i |a_{n,i}|=1$ in the statement of Theorem 4·7 cannot be removed.

In addition, the main result in [ 27 ] states that Theorem 4·2, specialised to the case $\mathcal{I}=\textrm{Fin}$ and $k=1$ , can be further strengtened so that the set $$\{ x \in (0,1]\;:\;\Gamma _b^1(x,{\rm{Fin}},A) \supseteq \Delta _b^1$$ for all $$b \ge 2$$ and all regular A} is comeager. Taking into account the argument in the proof of Theorem 4·7, this would imply that the set

(7) \begin{equation}\{x \in (0,1]\;:\; \Gamma_b^1 (x,\textrm{Fin},A)= \Delta_{b}^1 \text{ for all }b\ge 2 \text{ and all nonnegative regular } A\}\end{equation}

should be comeager. However, this is false as it is shown in the next example.

Example 4·10. For each $y \in (0,1]$ , let $(e_{y,k}\;:\; k\ge 1)$ be the increasing enumeration of the infinite set $\{n \in \textbf{N}\;:\; d_{2,n}(y)=1\}$ . Then, let $\mathscr{A}=\{A_y\;:\; y \in (0,1]\}$ be family of matrices $A_y=\left(a^{(y)}_{n,i}\right)$ with entries in $\{0,1\}$ so that $a^{(y)}_{n,i}=1$ if and only if $e_{y,n}=i$ for all $y \in (0,1]$ and all $n,i \in \textbf{N}$ . Then each $A_y$ is a nonnegative regular matrix. It follows, for each ideal $\mathcal{I}$ ,

\begin{equation*}\{x \in (0,1]\;:\; \Gamma_2^1 (x,\mathcal{I},A)=\Delta_{2}^1\text{ for all }A \in \mathscr{A} \}=\emptyset.\end{equation*}

Indeed, for each $x \in (0,1]$ , the sequence ${\boldsymbol{\pi}}_2^1(x)=({\boldsymbol{\pi}}_{2,n}^1(x)\;:\;n\ge 1)$ has an accumulation point ${\boldsymbol{\eta}} \in \Delta_2^1$ . Hence there exists a subsequence $({\boldsymbol{\pi}}_{2,n_k}^1(x)\;:\;k\ge 1)$ which is convergent to ${\boldsymbol{\eta}}$ . Equivalently, $\lim A_y{\boldsymbol{\pi}}_2^1(x)={\boldsymbol{\eta}}$ , where $y\in (0,1]$ is defined such that $e_{y,k}=n_k$ for all $k \in \textbf{N}$ . Therefore $\{{\boldsymbol{\eta}}\}=\Gamma_2^1 (x,\mathcal{I},A_y)\neq \Delta_{2}^1$ . in particular, the set defined in (7) is empty.

Lastly, the analogues of Theorem 4·2 and Theorem 4·7 hold for $\mathcal{I}$ -limit points, if $\mathcal{I}$ is an $F_\sigma$ -ideal or an analytic P-ideal. Indeed, denoting with $\Lambda_b^k(x,\mathcal{I},A)$ the set of $\mathcal{I}$ -limit points of the sequence $(A_n{\boldsymbol{\pi}}_b^k(x)\;:\; n\ge 1)$ , we obtain:

Theorem 4·11. Let A be a regular matrix and let $\mathcal{I}$ be an $F_\sigma$ -ideal or an analytic P-ideal. Then the set $\{x \in (0,1]\;:\; \Lambda_b^k (x,\mathcal{I},A)\supseteq \Delta_{b}^k \text{ for all }b\ge 2, k\ge 1\}$ is comeager.

Moreover, the set $\{x \in (0,1]\;:\; \Lambda_b^k (x,\mathcal{I},A)=\Delta_{b}^k \text{ for all }b\ge 2, k\ge 1\}$ is comeager if, in addition, A satisfies $\lim_n\sum_i |a_{n,i}|=1$ .

Proof. The first part goes along the same lines of the proof of Theorem 2·3. Here, we replace ${\boldsymbol{\pi}}_b^k(x)$ with $(A_n{\boldsymbol{\pi}}_b^k(x)\;:\; n\ge 1)$ and using the chain of inequalities (4): more precisely, we consider $j^{\prime\prime}\in \textbf{N}$ such that $\varphi(\{n \in [j^\prime, j^{\prime\prime}]\;:\; \|A_n {\boldsymbol{\pi}}^k_b(x^\prime)-{\boldsymbol{\eta}}\|\le \; {1}/{2t}\})\ge \; {1}/{2},$ and, taking into considering (4), we define $V \;:\!=\; \{x \in (0,1]\;:\; d_{b,i}(x)=d_{b,i}(x^\star) \text{ for all }i=1,\ldots,k+j^{\prime\prime\prime}\}$ , where $j^{\prime\prime\prime}$ is a sufficiently large integer such that $\sum_{i>j^{\prime\prime\prime}}|a_{n,i}|\le \; {1}/{2t}$ for all $n \in [j^\prime, j^{\prime\prime}]$ .

The second part follows, as in Theorem 4·7, by the fact that every accumulation point of $(A_n{\boldsymbol{\pi}}_b^k(x)\;:\; n\ge 1)$ belongs to $\Delta_b^k$ .