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.


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 x are normal from a measure theoretic viewpoint, see e.g.[5] for history and generalizations.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 as n → ∞, where s 0 s and ss k stand for the concatened strings (indeed, the above identity is obtained by a double counting of the occurrences of the string s as the occurrences of all possible strings ss k ; or, equivalently, as the occurrences of all possible strings s 0 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 L k b (x) of accumulation points of the sequence of vectors and Olsen proved in [23] that the subset of nonnormal numbers with maximal set of accumulation points is topologically large: First, we strenghten Theorem 1.1 by showing that the set of accumulation points L k b (x) can be replaced by the much smaller subset of accumulation points η such that every neighborhood of η contains "sufficiently many" elements of the sequence, where "sufficiently many" is meant with respect to a suitable ideal I of subsets of the positive integers N; see Theorem 2.1.Hence, Theorem 1.1 corresponds to the case where I is 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 η 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.

Main results
An ideal I ⊆ P(N) is a family closed under finite union and subsets.It is also assumed that I contains 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 G δ -subsets of P(N), F σ -ideals, meager ideals, analytic ideals, etc.In addition, we say that I is a P-ideal if it is σ-directed modulo finite sets, i.e., for each sequence (S n ) of sets in I there exists S ∈ I such that S n \ S is finite for all n ∈ N. Lastly, we denote by Z the ideal of asymptotic density zero sets, i.e., where d ⋆ (S) := lim sup 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 η ∈ X is an I-cluster point of x if {n ∈ N : x n ∈ U} / ∈ I for all open neighborhoods U of 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 Γ k b (x, I) be the set of I-cluster points of the sequence (π k b,n (x) : n ≥ 1).Theorem 2.1.The set {x ∈ (0, 1] : The class of meager ideals is really broad.Indeed, it contains Fin, Z, the summable ideal {S ⊆ N : n∈S 1/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 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 I. Indeed, if I is 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 ) is a singleton.On a similar direction, if x = (x n ) is a sequence taking values in a topological vector space X, then η ∈ X is an I-limit point of x if there exists a subsequence (x n k ) such that lim k x n k = η and N \ {n 1 , n 2 , . ..} ∈ 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 Λ k b (x, I) be the set of I-limit points of the sequence (π k b,n (x) : n ≥ 1).The analogue of Theorem 2.1 for I-limit points follows.
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 2.3 provides a further improvement on Theorem 2.1 for the subfamily of analytic P-ideals.
It is remarkable that there exist F σ -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 characterized 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 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 ∈ (0, 1] such that, for all b ≥ 2 and k ≥ 1, every vector in ∆ k b is a statistical limit point of the sequence (π k 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 2.3 can be extended for all F σδ -ideals including, in particular, the ideal I generated by the upper Banach density (which is known to not be a P-ideal, see e.g.[12, p.299]).

Proofs of the main results
Proof of Theorem 2.1.Let I be a meager ideal on N. It follows by Talagrand's characterization of meager ideals [28,Theorem 21] that it is possible to define a partition {I 1 , I 2 , . ..} of N into nonempty finite subsets such that S / ∈ I whenever I n ⊆ S for infinitely many n.Moreover, we can assume without loss of generality that max I n < min I n+1 for all n ∈ N.
The claimed set can be rewritten as b≥2 k≥1 X k b , where Since the family of meager subsets of (0, 1] is a σ-ideal, it is enough to show that the complement of each X k b is meager.To this aim, fix b ≥ 2 and k ≥ 1 and denote by 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 ∈ N and a nonempty relatively open set G ⊆ (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 s = s 1 • • • s j ∈ S j b such that x ∈ G whenever d b,i (x) = s i for all i = 1, . . ., j.Now, pick x ⋆ ∈ (0, 1] such that lim n π k b,n (x ⋆ ) = η t , which exists by [22,Theorem 1].In addition, we can assume without loss of generality that d b,i (x ⋆ ) = s i for all i = 1, . . ., j.Since π k b,n (x ⋆ ) is convergent to η t , there exists q ≥ p + j such that π k b,n (x ⋆ ) − η t < 1 /m for all n ≥ min I q .Define V := {x ∈ (0, 1] : d b,i (x) = d b,i (x ⋆ ) for all i = 1, . . ., max I q + k} and note that V ⊆ G because d b,i (x) = s i for all i ≤ j and x ∈ V , and V ∩ S t,m,p = ∅ because, for each x ∈ V , the required property is not satisfied for this choice of q since π k b,n (x) = π k b,n (x ⋆ ) for all n ≤ max I q .Clearly, V has nonempty interior, hence it is possible to choose such U ⊆ 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 characterization of analytic P-ideals.A lower semicontinuous submeasure (in short, lscsm) is a monotone subadditive function ϕ :  Then, we assume hereafter that I is an analytic P-ideal generated by a lscsm ϕ as in (3).Fix integers b ≥ 2 and k ≥ 1, and define the function 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 b and notice that 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 ⊆ (0, 1], s ∈ S j b , and x ⋆ ∈ (0, 1] as in the proof of Theorem 2.1.Considering that • ϕ is invariant under finite sets, it follows that where j ′ := j + h.Since ϕ is lower semicontinuous, there exists an integer j ′′ > j ′ such that Finally, let E be a countable dense subset of ∆ k b .Considering that X := {x ∈ (0, 1] : , it follows that the set X is comeager.However, considering that In particular, the claimed set contains X, which is comeager.This concludes the proof.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 ∈ N) be a regular matrix, that is, an infinite real-valued matrix such that, if z = (z n ) is a R d -valued sequence convergent to η, then A n z := i a n,i z i exists for all n ∈ N and lim n A n z = η, see e.g.[9,Chapter 4].Then, for each x ∈ (0, 1] and integers b ≥ 2 and k ≥ 1, let Γ k b (x, I, A) be the set of I-cluster points of the sequence of vectors Proof.Fix a regular matrix A = (a n,i ) and a meager ideal I.The proof goes along the same lines as the proof of Theorem 2.1, replacing the definition of S t,m,p with S ′ t,m,p := {x ∈ (0, 1] : ∀q ≥ p, ∃n ∈ I q , A n π k b (x) − η t ≥ 1 /m}.Recall that, thanks to the classical Silverman-Toeplitz characterization of regular matrices, see e.g.[9, Theorem 4.1, II] or [8], we have that sup n i |a n,i | < ∞.Since lim n π k b,n (x ⋆ ) = η t , it follows that there exist sufficiently large integers q ≥ p + j and j A ≥ j such that, if d b,i (x) = d b,i (x ⋆ ) for all i = 1, . . ., j A + k, then for all n ∈ I q .We conclude analogously that S ′ t,m,p is 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 = (x n ) with values in R k , 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 L x , where co S is the convex hull of S ⊆ R k and L x is the set of accumulation points of x.The ideal version of the Knopp core has been studied in [18,16].The classical Knopp theorem states that, if k = 2 and A is a nonnegative regular matrix, then for all bounded sequences x, where Ax = (A n x : n ≥ 1), see [17, p. 115]; cf.[9,Chapter 6] for a textbook exposition.A generalization 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. .Similarly, let Q(a) be the vector in K which minimizes its distance with a.Then, notice that, for all n, m ∈ N, we have ) by the continuity of d(•, K), it is sufficient to show that both d(A n x, K m ) and sup y∈Km d(y, K) are sufficiently small if n is sufficiently large and m is chosen properly.
To this aim, fix ε > 0 and choose m ∈ N such that sup y∈Km d(y, K) ≤ ε /2.Indeed, it is sufficient to choose m ∈ N such that d(x n , L x ) < ε /2 for all n ≥ m: indeed, in the opposite, the subsequence (x j ) j∈J , where J := {n ∈ N : d(x n , L x ) ≥ ε /2}, would be bounded and without any accumulation point, which is impossible.Now pick y ∈ K m so that y = j λ i j x i j for some strictly increasing sequence (i j ) of positive integers such that i 1 ≥ m and some real nonnegative sequence (λ i j ) with j λ i j = 1.It follows that Suppose for the moment that A has nonnegative entries.Since A is regular, we get lim n i a n,i = 1 and lim n i<m a n,i = 0 by the Silverman-Toeplitz characterization, hence lim n i≥m a n,i = 1 and there exists n 0 ∈ N such that i≥m a n,i ≥ 1 /2 for all n ≥ n 0 .Thus, for each n ≥ n 0 , we obtain that d Recalling that κ = sup n x n , it is easy to see that In addition, setting t n := i≥m a n,i / i a n,i ∈ [0, 1] for all n ≥ n 0 , we get where ⋆ i∈I stands for i∈I a n,i 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 ⋆ i≥m / i≥m a n,i ∈ K m .Since lim n i<m |a n,i | = 0, lim n t n = 1, and sup n i |a n,i | < ∞ by the regularity of A, it follows that all α n , β n , γ n are smaller than ε /6 if n is sufficiently large.Therefore d(A n x, K) ≤ ε and, since ε is arbitrary, we conclude that η = lim n A n x ∈ K.
Lastly, suppose that A is a regular matrix such that lim n 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 ∈ N. Considering that and that lim n i |a n,i − |a n,i || = 0 because lim n i a n,i = lim n i |a n,i | = 1, we conclude that d(A n x, K m ) ≤ 2ε whenever n is sufficiently large.The claim follows as before.
The following corollary is immediate: Corollary 4.4.Let x = (x n ) be a bounded sequence taking values in 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 ∈ 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 ∈ N. Then A is regular and lim Ax = 3 / ∈ {−1, 1} = K-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 K-core(x) with the closure of co L x (this coincides in the case that X = R k ) and assume that the sequence x is contained in a compact set (so that K-core(x) is also nonempty).
With these premises, we can strenghten Theorem 4.2 as follows.Proof.Let us suppose that A = (a n,i ) is nonnegative regular matrix, i.e., a n,i ≥ 0 for 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 (A n π k b (x) : n ≥ 1) is contained in the convex hull of the set of accumulation points of (π k b,n (x) : n ≥ 1), which is in turn contained into ∆ k b .This follows by Proposition 4.3.
with each digit d b,n (x) ∈ {0, 1, . . ., b − 1}, where b ≥ 2 is a given integer.Then, for each string s = s 1 • • • s k with digits s j ∈ {0, 1, . . ., b − 1} and each n ≥ 1, write π b,s,n (x) for the proportion of strings s in the b-adic expansion of x which start at some position ≤ n, i.e., π b,s,n (x) := #{i ∈ {1, . . ., n} : d b,i+j−1 (x) = s j for all j = 1, . . ., k} n .In addition, let S k b be the set of all possible strings s = s 1 • • • s k in base b of length k, hence #S k b = b k , and denote by π k b,n (x) the vector (π b,s,n (x) : s ∈ S k b ).Of course, π k b,n (x) belongs to the (b k − 1)-dimensional simplex for each n.However, the components of π k b,n (x) satisfy an additional requirement: if k ≥ 2 and s = s 1 • • • s k−1 is a string in S k−1 b , then π b,s,n (x) = s k π b,ss k ,n (x) = s 0 π b,s 0 s,n (x) + O (1/n)

4 . Applications 4 . 1 .Proposition 4 . 1 .
Hausdorff and packing dimensions.We refer to[10, Chapter 3]  for the definitions of the Hausdorff dimension and the packing dimension.The sets defined in Theorem 2.1 and Theorem 2.3 have Hausdorff dimension 0 and packing dimension 1.

Proposition 4 . 3 .
Let x = (x n ) be a bounded sequence taking values in R k , and fix a regular matrix A such that lim n i |a n,i | = 1.Then inclusion (5) holds.Proof.Define κ := sup n x n and let η be 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 K m be the closure of co{x m , x m+1 , . ..}, hence K ⊆ K m .Define d(a, C) := min b∈C a − b for all a ∈ R k and nonempty compact sets C ⊆ R k .In addition, for each m ∈ N, let Q m (a) ∈ K m be the unique vector such that d(a, K m ) = a − Q m (a)

Theorem 4 . 7 .
The set {x ∈ (0, 1] : Γ k b (x, I, A) = ∆ k b for all b ≥ 2, k ≥ 1} is comeager, provided that I is a meager ideal and A is a regular matrix such that lim n i |a n,i | = 1.
t} ϕ where • stands for the Euclidean norm on R b k .It follows by [2, Lemma 2.1] that every section u(x, •) is upper semicontinuous, so that the set