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Math. Proc. Camb. Phil. Soc.: page 1 of 11 1
doi:10.1017/S0305004122000469
Most numbers are not normal
BYANDREA AVENI
Department of Statistical Science, Duke University, 214 Old Chemistry, Durham, NC
27708-0251, U.S.A.
e-mail:andrea.aveni@duke.edu
AND PAOLO LEONETTI
Department of Economics, Università degli Studi dell’Insubria, via Monte Generoso 71,
Varese 21100, Italy.
e-mail:leonetti.paolo@gmail.com
(Received 25 January 2021; revised 07 November 2022; accepted 31 October 2022)
Abstract
We show, from a topological viewpoint, that most numbers are not normal in a strong
sense. More precisely, the set of numbers x∈(0,1] with the following property is comea-
ger: for all integers b≥2 and k≥1, the sequence of vectors made by the frequencies of all
possibile strings of length kin the b-adic representation of xhas a maximal subset of accu-
mulation 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.
2020 Mathematics Subject Classification: 11K16 (Primary); 40A35, 11A63 (Secondary)
1. Introduction
A real number x∈(0, 1] is normal if, informally, for each base b≥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 xare 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∈(0, 1], denote its unique nonterminating b-adic expansion by
x=n≥1
db,n(x)
bn,(1)
with each digit db,n(x)∈{0, 1, ...,b−1}, where b≥2 is a given integer. Then, for each
string s=s1···skwith digits sj∈{0, 1, ...,b−1}and each n≥1, write πb,s,n(x) for the
C
The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society. This is an Open Access
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2A. AVENI AND P. LEONETTI
proportion of strings sin the b-adic expansion of xwhich start at some position ≤n, i.e.,
πb,s,n(x):=#{i∈{1, ...,n}:db,i+j−1(x)=sjfor all j=1, ...,k}
n.
In addition, let Sk
bbe the set of all possible strings s=s1···skin base bof length k, hence
#Sk
b=bk, and denote by πk
b,n(x) the vector (πb,s,n(x):s∈Sk
b). Of course, πk
b,n(x) belongs to
the (bk−1)-dimensional simplex for each n. However, the components of πk
b,n(x) satisfy an
additional requirement: if k≥2 and s=s1···sk−1is a string in Sk−1
b, then
πb,s,n(x)=sk
πb,ssk,n(x)=s0πb,s0s,n(x)+O(1/n)as n→∞,
where s0sand sskstand for the concatened strings (indeed, the above identity is obtained
by a double counting of the occurrences of the string sas the occurrences of all possible
strings ssk; or, equivalently, as the occurrences of all possible strings s0s, 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 Lk
b(x) of accumulation points of the sequence of vectors (πk
b,n(x):n≥1)
is contained in k
b, where
k
b:=(ps)s∈Sk
b∈Rbk:sps=1, ps≥0 for all s∈Sk
b,
and s0ps0s=sk
psskfor all s∈Sk−1
b.
Then xis said to be normal if
∀b≥2, ∀k≥1, ∀s∈Sk
b, lim
n→∞ πb,s,n(x)=1/bk.
Hence, if xis normal, then Lk
b(x)={(1/bk,...,1/bk)}. 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∈(0, 1] : Lk
b(x)=k
bfor all b ≥2, k≥1}is comeager.
First, we strenghten Theorem 1·1by showing that the set of accumulation points Lk
b(x)
can be replaced by the much smaller subset of accumulation points ηsuch that every neigh-
bourhood of ηcontains “sufficiently many” elements of the sequence, where “sufficiently
many” is meant with respect to a suitable ideal Iof subsets of the positive integers N; see
Theorem 2·1. Hence, Theorem 1·1corresponds to the case where Iis the family of finite
sets.
Then, for certain ideals I(including the case of the family of asymptotic density zero
sets), we even strenghten the latter result by showing that each accumulation point ηcan 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 I⊆P(N) is a family closed under finite union and subsets. It is also assumed
that Icontains the family of finite sets Fin and it is different from P(N). Every subset of
P(N) is endowed with the relative Cantor-space topology. In particular, we may speak about
Gδ-subsets of P(N), Fσ-ideals, meager ideals, analytic ideals, etc. In addition, we say that
https://doi.org/10.1017/S0305004122000469 Published online by Cambridge University Press
Most numbers are not normal 3
Iis a P-ideal if it is σ-directed modulo finite sets, i.e., for each sequence (Sn) of sets in I
there exists S∈Isuch that Sn\Sis finite for all n∈N. Lastly, we denote by Zthe ideal of
asymptotic density zero sets, i.e.,
Z=S⊆N:d(S)=0,(2)
where d(S):=lim supn1
n#(S∩[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=(xn) be a sequence taking values in a topological vector space X. Then we say that
η∈Xis an I-cluster point of xif {n∈N:xn∈U}/∈Ifor all open neighbourhoods Uof η.
Note that Fin-cluster points are the ordinary accumulation points. Usually Z-cluster points
are referred to as statistical cluster points, see e.g. [13]. It is worth noting that 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∈(0, 1] and for all integers b≥2 and k≥1, let k
b(x,I)
be the set of I-cluster points of the sequence (πk
b,n(x):n≥1).
THEOREM 2·1. The set {x∈(0, 1] : k
b(x,I)=k
bfor all b ≥2, k≥1}is comeager, pro-
vided that Iis a meager ideal.
The class of meager ideals is really broad. Indeed, it contains Fin, Z, the summable ideal
{S⊆N:n∈S1/n<∞}, the ideal generated by the upper Banach density, the analytic P-
ideals, the Fubini sum Fin ×Fin, the random graph ideal, etc.; cf. e.g. [3,14]. Note that
k
b(x,I)=Lk
b(x)ifI=Fin. Therefore Theorem 2·1significantly strenghtens Theorem 1·1.
Remark 2·2. It is not difficult to see that Theorem 2·1does not hold without any restriction
on I. Indeed, if Iis a maximal ideal (i.e., the complement of a free ultrafilter on N), then
for each x∈(0, 1] and all integers b≥2, k≥1, we have that the sequence (πk
b,n(x):n≥1) is
bounded, hence it is I-convergent so that k
b(x,I) is a singleton.
On a similar direction, if x=(xn) is a sequence taking values in a topological vector space
X, then η∈Xis an I-limit point of xif there exists a subsequence (xnk) such that limkxnk=η
and N\{n1,n2,...}∈I. Usually Z-limit points are referred to as statistical limit points, see
e.g. [13]. Similarly, for each x∈(0,1] and for all integers b≥2 and k≥1, let k
b(x,I)be
the set of I-limit points of the sequence (πk
b,n(x):n≥1). The analogue of Theorem 2·1for
I-limit points follows.
THEOREM 2·3. The set {x∈(0, 1] : k
b(x,I)=k
bfor all b ≥2, k≥1}is comeager, pro-
vided that Iis an analytic P-ideal or an Fσ-ideal.
It is known that every I-limit point is always an I-cluster point, however they can be
highly different, as it is shown in [2, theorem 3·1]. This implies that Theorem 2·3provides
a further improvement on Theorem 2·1for the subfamily of analytic P-ideals.
It is remarkable that there exist Fσ-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 charac-
terised with the aid of lower semicontinuous submeasures, cf. Section 3. The results in
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4A. AVENI AND P. LEONETTI
[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 Zdefined 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 ∈(0, 1] such that, for all b ≥2and k ≥1, every vector in k
b
is a statistical limit point of the sequence (πk
b,n(x):n≥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 2·3can
be extended for all Fσδ-ideals including, in particular, the ideal Igenerated 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 Ibe a meager ideal on N. It follows by Talagrand’s character-
isation of meager ideals [28, theorem 21] that it is possible to define a partition {I1,I2,...}
of Ninto nonempty finite subsets such that S/∈Iwhenever In⊆Sfor infinitely many n.
Moreover, we can assume without loss of generality that maxIn<min In+1for all n∈N.
The claimed set can be rewritten as b≥2k≥1Xk
b, where Xk
b:={x∈(0, 1] : k
b(x,I)=
k
b}. Since the family of meager subsets of (0,1] is a σ-ideal, it is enough to show that
the complement of each Xk
bis meager. To this aim, fix b≥2 and k≥1 and denote by ·
the Euclidean norm on Rbk. Considering that {η1,η2,...}:=k
b∩Qbkis a countable dense
subset of k
band that k
b(x,I) is a closed subset of k
bby [19, lemma 3·1(iv)], it follows
that
(0, 1] \Xk
b=t≥1{x∈(0, 1] : ηt/∈k
b(x,I)}
=t≥1{x∈(0, 1] : ∃ε>0, {n∈N:πk
b,n(x)−ηt<ε}∈I}
⊆t,p,m≥1{x∈(0, 1] : ∀q≥p,∃n∈Iq,πk
b,n(x)−ηt≥ 1/m}.
Denote by St,m,pthe set in the latter union. Thus it is sufficient to show that each St,p,mis
nowhere dense. To this aim, fix t,p,m∈Nand a nonempty relatively open set G⊆(0, 1].
We claim there exists a nonempty open set Ucontained in Gand disjoint from St,p,m. Since
Gis nonempty and open in (0, 1], there exists a string ˜s=s1···sj∈Sj
bsuch that x∈G
whenever db,i(x)=sifor all i=1, ...,j. Now, pick x∈(0, 1] such that limnπk
b,n(x)=ηt,
which exists by [22, theorem 1]. In addition, we can assume without loss of generality
that db,i(x)=sifor all i=1, ...,j. Since πk
b,n(x) is convergent to ηt, there exists q≥
p+jsuch that πk
b,n(x)−ηt<1/mfor all n≥min Iq. Define V:={x∈(0, 1] : db,i(x)=
db,i(x) for all i=1, ...,maxIq+k}and note that V⊆Gbecause db,i(x)=sifor all i≤j
and x∈V, and V∩St,m,p=∅because, for each x∈V, the required property is not satisfied
for this choice of qsince πk
b,n(x)=πk
b,n(x) for all n≤max Iq. Clearly, Vhas nonempty
interior, hence it is possible to choose such U⊆V.
This proves that each St,m,pis nowhere dense, concluding the proof.
https://doi.org/10.1017/S0305004122000469 Published online by Cambridge University Press
Most numbers are not normal 5
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 ϕ:P(N)→[0, ∞] such that ϕ(∅)=0, ϕ({n})<∞, and
ϕ(A)=limmϕ(A∩[1, m]) for all A⊆Nand n∈N. It follows by [26, theorem 3·1] that an
ideal Iis an analytic P-ideal if and only if there exists a lscsm ϕsuch that
I={A⊆N:Aϕ=0},Nϕ=1, and ϕ(N)<∞.(3)
Here, Aϕ:=limnϕ(A\[1, n]) for all A⊆N. Note that Aϕ=Bϕwhenever the sym-
metric difference ABis finite, cf. [11, lemma 1·3·3(b)]. Easy examples of lscsms are
ϕ(A):=#Aor ϕ(A):=supn(1/n)#(A∩[1, n]) for all A⊆Nwhich lead, respectively, to the
ideals Fin and Zthrough the representation (3).
Proof of Theorem 2·3. First, let us suppose that Iis an Fσ-ideal. We obtain by [2,
theorem 2·3] that k
b(x,I)=k
b(x,I) for each b≥2, k≥1, and x∈(0, 1]. Therefore the
claim follows by Theorem 2·1.
Then, we assume hereafter that Iis an analytic P-ideal generated by a lscsm ϕas in (3).
Fix integers b≥2 and k≥1, and define the function
u: (0, 1] ×k
b−→ R:(x,η)−→ lim
t→∞ {n∈N:πk
b,n(x)−η≤ 1/t}ϕ,
where · stands for the Euclidean norm on Rbk. It follows by [2, lemma 2·1] that every
section u(x,·) is upper semicontinuous, so that the set
k
b(x,I,q):={η∈k
b:u(x,η)≥q}
is closed for each x∈(0, 1] and q∈R.
At this point, we prove that, for each η∈k
b, the set X(η):={x∈(0, 1] : u(x,η)≥1/2}
is comeager. To this aim, fix η∈k
band notice that
(0, 1] \X(η)=t≥1{x∈(0, 1] : {n∈N:πk
b,n(x)−η≤ 1/t}ϕ<1/2}
=t≥1{x∈(0, 1] : lim
h→∞ ϕ({n≥h:πk
b,n(x)−η≤ 1/t})<1/2}
=t,h≥1{x∈(0, 1] : ϕ({n≥h:πk
b,n(x)−η≤ 1/t})<1/2}.
Denoting by Yt,hthe inner set above, it is sufficient to show that each Yt,his nowhere dense.
Hence, fix G⊆(0, 1], ˜s∈Sj
b, and x∈(0, 1] as in the proof of Theorem 2·1. Considering that
·ϕis invariant under finite sets, it follows that
ϕ({n≥j:πk
b,n(x)−η≤ 1/t})≥{n≥j:πk
b,n(x)−η≤ 1/t}ϕ=u(x,η)=1,
where j:=j+h. Since ϕis lower semicontinuous, there exists an integer j >jsuch that
ϕ({n∈[j,j]:πk
b,n(x)−η≤ 1/t})≥1/2.
Define V:={x∈(0, 1] : db,i(x)=db,i(x) for all i=1, ...,j}. Similarly, note that V⊆G
because db,i(x)=sifor all i≤jand x∈V, and V∩Yt,h=∅ because ϕ({n≥h:πk
b,n(x)−
η≤ 1/t}) is at least ϕ({n∈[j,j]:πk
b,n(x)−η≤ 1/t})≥1/2 for all x∈V. Since Vhas
nonempty interior, it is possible to choose U⊆Vwith the required property.
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6A. AVENI AND P. LEONETTI
Finally, let Ebe a countable dense subset of k
b. Considering that X:={x∈(0, 1] : E⊆
k
b(x,I,1/2)}is equal to η∈EX(η), it follows that the set Xis comeager. However,
considering that
k
b(x,I)=q>0k
b(x,I,q)
by [2, theorem 2·2] and that k
b(x,I,1/2) is a closed subset such that E⊆k
b(x,I,1/2) ⊆
k
b(x,I)⊆k
bfor all x∈X, we obtain that k
b(x,I,1/2) =k
b(x,I)=k
bfor all x∈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 2·1and Theorem 2·3have Hausdorff
dimension 0and 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=(an,i:n,i∈N)be
aregular matrix, that is, an infinite real-valued matrix such that, if z=(zn)isaRd-valued
sequence convergent to η, then Anz:=ian,iziexists for all n∈Nand limnAnz=η, see
e.g. [9, chapter 4]. Then, for each x∈(0, 1] and integers b≥2 and k≥1, let k
b(x,I,A)be
the set of I-cluster points of the sequence of vectors Anπk
b(x):n≥1, where πk
b(x)isthe
sequence (πk
b,n(x):n≥1).
In particular, k
b(x,I,A)=k
b(x,I)ifAis the infinite identity matrix.
THEOREM 4·2. The set {x∈(0, 1] : k
b(x,I,A)⊇k
bfor all b ≥2, k≥1}is comeager,
provided that Iis a meager ideal and A is a regular matrix.
Proof. Fix a regular matrix A=(an,i) and a meager ideal I. The proof goes along the
same lines as the proof of Theorem 2·1, replacing the definition of St,m,pwith
S
t,m,p:={x∈(0, 1] : ∀q≥p,∃n∈Iq,Anπk
b(x)−ηt≥ 1/m}.
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 supni|an,i|<∞. Since limnπk
b,n(x)=ηt,
it follows that there exist sufficiently large integers q≥p+jand jA≥jsuch that, if db,i(x)=
db,i(x) for all i=1, ...,jA+k, then
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Most numbers are not normal 7
Anπk
b(x)−ηt≤Anπk
b,(x)−ηt+
ian,i(πk
b,i(x)−πk
b,i(x))
≤Anπk
b(x)−ηt+i|an,i|πk
b,i(x)−πk
b,i(x)
≤Anπk
b(x)−ηt+i>jA
|an,i|<1
m(4)
for all n∈Iq. We conclude analogously that S
t,m,pis nowhere dense.
The main result in [27] corresponds to the case I=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 x=(xn) with values in Rk, let K-core(x) be the Knopp core of x, that is,
the convex hull of the set of accumulation points of x. In other words, K-core(x)=co Lx,
where co Sis the convex hull of S⊆Rkand Lxis the set of accumulation points of x. The
ideal version of the Knopp core has been studied in [16,18]. The classical Knopp theorem
states that, if k=2 and Ais a nonnegative regular matrix, then
K-core(Ax)⊆K-core(x)(5)
for all bounded sequences x, where Ax=(Anx:n≥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∈N.
PROPOSITION 4·3. Let x=(xn)be a bounded sequence taking values in Rk, and fix a
regular matrix A such that limni|an,i|=1. Then inclusion (5)holds.
Proof. Define κ:=supnxnand let ηbe an accumulation point of Ax. It is sufficient
to show that η∈K:=K-core(x). Possibly deleting some rows of A, we can assume without
loss of generality that lim Ax=η. For each m∈N, let Kmbe the closure of co{xm,xm+1,...},
hence K⊆Km. Define d(a,C):=minb∈Ca−bfor all a∈Rkand nonempty compact
sets C⊆Rk. In addition, for each m∈N, let Qm(a)∈Kmbe the unique vector such that
d(a,Km)=a−Qm(a). Similarly, let Q(a) be the vector in Kwhich minimizes its distance
with a. Then, notice that, for all n,m∈N,wehave
d(Anx,K)≤infb∈Kinfc∈Rk(Anx−c+c−b)
≤infc∈Kminfb∈K(Anx−c+c−b)
≤infc∈KmAnx−c+supy∈Kminfb∈Ky−b
=d(Anx,Km)+supy∈Kmd(y,K).
Since d(η,K)=limnd(Anx,K) by the continuity of d(·,K), it is sufficient to show that both
d(Anx,Km) and supy∈Kmd(y,K) are sufficiently small if nis sufficiently large and mis chosen
properly.
To this aim, fix ε>0 and choose m∈Nsuch that supy∈Kmd(y,K)≤ε/2. Indeed, it is
sufficient to choose m∈Nsuch that d(xn,L
x)<ε/2 for all n≥m: indeed, in the opposite,
the subsequence (xj)j∈J, where J:={n∈N:d(xn,L
x)≥ε/2}, would be bounded and with-
out any accumulation point, which is impossible. Now pick y∈Kmso that y=jλijxijfor
some strictly increasing sequence (ij) of positive integers such that i1≥mand some real
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8A. AVENI AND P. LEONETTI
nonnegative sequence (λij) with jλij=1. It follows that
d(y,K)≤
y−jλijQ(xij)
≤jλij
xij−Q(xij)
≤jλijd(xij,Lx)≤ε
2.
Suppose for the moment that Ahas nonnegative entries. Since Ais regular, we get
limnian,i=1 and limni<man,i=0 by the Silverman–Toeplitz characterisation, hence
limni≥man,i=1 and there exists n0∈Nsuch that i≥man,i≥1/2 for all n≥n0. Thus,
for each n≥n0, we obtain that d(Anx,Km)=Anx−Qm(Anx)≤αn+βn+γn, where
αn:=
Anx−Anx
ian,i
,βn:=
Anx
ian,i
−QmAnx
ian,i
and
γn:=
QmAnx
ian,i−Qm(Anx)
.
Recalling that κ=supnxn, it is easy to see that
γn≤αn≤κi|an,i|·1−1
ian,i.
In addition, setting tn:=i≥man,i/ian,i∈[0, 1] for all n≥n0, we get
βn≤
i
ian,i
−
i≥m
i≥man,i
=1
i≥man,iian,i
i≥man,i
i<m+
i≥m−ian,i
i≥m
=1
i≥man,i
tn
i<m+(1 −tn)
i≥m
≤2κtni<m|an,i|+(1 −tn)i|an,i|,(6)
where
i∈Istands for i∈Ian,ixi. Note that the hypothesis that the entries of Aare non-
negative has been used only in the first line of (6), so that
i≥m/i≥man,i∈Km. Since
limni<m|an,i|=0, limntn=1, and supni|an,i|<∞by the regularity of A, it follows
that all αn,βn,γnare smaller than ε/6ifnis sufficiently large. Therefore d(Anx,K)≤εand,
since εis arbitrary, we conclude that η=limnAnx∈K.
Lastly, suppose that Ais a regular matrix such that limni|an,i|=1 and let B=(bn,i)be
the nonnegative regular matrix defined by bn,i=|an,i|for all n,i∈N. Considering that
d(Anx,Km)≤Anx−Bnx+d(Bnx,Km)≤κi|an,i−|an,i|| + ε,
and that limni|an,i−|an,i|| = 0 because limnian,i=limni|an,i|=1, we conclude
that d(Anx,Km)≤2εwhenever nis sufficiently large. The claim follows as before.
The following corollary is immediate:
COROLLARY 4·4. Let x=(xn)be a bounded sequence taking values in Rk, and fix a
nonnegative regular matrix A. Then inclusion (5)holds.
https://doi.org/10.1017/S0305004122000469 Published online by Cambridge University Press
Most numbers are not normal 9
Remark 4·5. Inclusion (5) fails for an arbitrary regular matrix: indeed, let A=(an,i) be the
matrix defined by an,2n=2, an,2n−1=−1 for all n∈N, and an,i=0 otherwise. Set also
k=1 and let xbe the sequence such that xn=(−1)nfor all n∈N. Then Ais regular and
lim Ax =3/∈{−1, 1}=K-core(x).
Remark 4·6. Proposition 4·3keeps holding on a (possibly infinite dimensional) Hilbert
space Xwith the following provisoes: replace the definition of K-core(x) with the closure of
co Lx(this coincides in the case that X=Rk) and assume that the sequence xis contained in
a compact set (so that K-core(x) is also nonempty).
With these premises, we can strenghten Theorem 4·2as follows.
THEOREM 4·7. The set {x∈(0, 1] : k
b(x,I,A)=k
bfor all b ≥2, k≥1}is comeager,
provided that Iis a meager ideal and A is a regular matrix such that limni|an,i|=1.
Proof. Let us suppose that A=(an,i) is nonnegative regular matrix, i.e., an,i≥0for
all n,i∈N, and fix a meager ideal I, a real x∈(0, 1], and integers b≥2, k≥1. Thanks
to Theorem 4·2, it is sufficient to show that every accumulation point of the sequence
(Anπk
b(x):n≥1) is contained in the convex hull of the set of accumulation points of
(πk
b,n(x):n≥1), which is in turn contained into k
b. This follows by Proposition 4·3.
Since the family of meager sets is a σ-ideal, the following is immediate by Theorem 4·7.
COROLLARY 4·8. Let Abe a countable family of regular matrices such that
limni|an,i|=1. Then the set {x∈(0, 1] : k
b(x,I,A)=k
bfor all b ≥2, k≥1, and all A ∈
A}is comeager, provided that Iis a meager ideal.
It is worth to remark that the main result [15] is obtained as an instance of Corollary 4·8,
letting Abe the set of iterates of the Cesàro matrix (note that they are nonnegative regular
matrices), and setting k=1 and I=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 limni|an,i|=1 for the entries of the regular matrix
in Theorem 4·7cannot be removed.
Example 4·9. Let A=(an,i) be the matrix such that an,(2n−1)!=−1 and an,(2n)!=2 for all
n∈N, and an,i=0 otherwise. It is easily seen that A is regular. Then, set b=2, k=1, and
I=Fin. We claim that the set of all x∈(0, 1] such that 2 is an accumulation point of the
sequence π2,1(x)=(π2,1,n(x):n≥1) is comeager. Indeed, its complement can be rewritten
as m,pSm,p, where
Sm,p:={x∈(0, 1] : |Anπ2,1(x)−2|≥ 1/mfor all n≥p}.
Let x∈(0, 1] such that d2,n(x)=1 if and only if (2i−1)!≤n<(2i)!for some i∈N. Then
it is easily seen that limnπ2,1,n(x)=2. Along the same lines of the proof of Theorem 4·2,it
follows that each Sm,pis meager. We conclude that {x∈(0, 1] : 1
2(x, Fin, A)=1
2}is meager,
which proves that the condition limni|an,i|=1 in the statement of Theorem 4·7cannot
be removed.
https://doi.org/10.1017/S0305004122000469 Published online by Cambridge University Press
10 A. AVENI AND P. LEONETTI
In addition, the main result in [27] states that Theorem 4·2, specialised to the case
I=Fin and k=1, can be further strengtened so that the set {x∈(0, 1] : 1
b(x, Fin, A)⊇1
b
for all b≥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
{x∈(0, 1] : 1
b(x,Fin,A)=1
bfor all b≥2 and all nonnegative regular A}(7)
should be comeager. However, this is false as it is shown in the next example.
Example 4·10. For each y∈(0, 1], let (ey,k:k≥1) be the increasing enumeration of the
infinite set {n∈N:d2,n(y)=1}. Then, let A={Ay:y∈(0, 1]}be family of matrices Ay=
a(y)
n,iwith entries in {0, 1}so that a(y)
n,i=1 if and only if ey,n=ifor all y∈(0, 1] and all
n,i∈N. Then each Ayis a nonnegative regular matrix. It follows, for each ideal I,
{x∈(0, 1] : 1
2(x,I,A)=1
2for all A∈A}=∅.
Indeed, for each x∈(0, 1], the sequence π1
2(x)=(π1
2,n(x):n≥1) has an accumulation
point η∈1
2. Hence there exists a subsequence (π1
2,nk(x):k≥1) which is convergent to
η. Equivalently, lim Ayπ1
2(x)=η, where y∈(0, 1] is defined such that ey,k=nkfor all k∈N.
Therefore {η}=1
2(x,I,Ay)= 1
2. in particular, the set defined in (7) is empty.
Lastly, the analogues of Theorem 4·2and Theorem 4·7hold for I-limit points, if Iis an
Fσ-ideal or an analytic P-ideal. Indeed, denoting with k
b(x,I,A) the set of I-limit points
of the sequence (Anπk
b(x):n≥1), we obtain:
THEOREM 4·11. Let A be a regular matrix and let Ibe an Fσ-ideal or an analytic
P-ideal. Then the set {x∈(0, 1] : k
b(x,I,A)⊇k
bfor all b ≥2, k≥1}is comeager.
Moreover, the set {x∈(0, 1] : k
b(x,I,A)=k
bfor all b ≥2, k≥1}is comeager if, in
addition, A satisfies limni|an,i|=1.
Proof. The first part goes along the same lines of the proof of Theorem 2·3. Here, we
replace πk
b(x) with (Anπk
b(x):n≥1) and using the chain of inequalities (4): more precisely,
we consider j ∈Nsuch that ϕ({n∈[j,j]:Anπk
b(x)−η≤ 1/2t})≥1/2, and, taking
into considering (4), we define V:={x∈(0, 1] : db,i(x)=db,i(x) for all i=1, ...,k+j},
where j is a sufficiently large integer such that i>j |an,i|≤ 1/2tfor all n∈[j,j].
The second part follows, as in Theorem 4·7, by the fact that every accumulation point of
(Anπk
b(x):n≥1) belongs to k
b.
Acknowledgments. P. Leonetti is grateful to PRIN 2017 (grant 2017CY2NCA) for
financial support.
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