We start by recalling the definition of natural density. For n, m ∈ ℕ with n < m, let [n, m] denote the set {n, n + 1, n + 2, …, m}. Let A ⊂ ℕ. Define

The numbers d̄(A) and d(A) are called the upper natural density and the lower natural density of A, respectively. If d̄(A) = d(A), then this common value is called the natural density of A and we denote it by d(A). Let ℐ_d be the family of all subsets of ℕ which have natural density 0. This ℐ_d is a proper nontrivial admissible ideal of subsets of ℕ. The notion of natural density was used by Fast [5] and Scoenberg [18] to define the notion of statistical convergence. Details of statistical convergence and later on, ideal convergence are thoroughly described in [1, 3, 4, 6, 7, 8, 10, 11, 12, 14, 15, 17, 19].

In [16] Osikiewicz had developed the ideas of finite and infinite splices. Let E₁, E₂, E₃, …, E_k, … be a partition of ℕ into countable number of sequences. Let y₁, y₂, y₃, …, y_k, … be distinct real numbers. Let (x_n) be such that

Then (x_n) is called an infinite-splice (In the same way Osikiewicz defined an finite splice taking finite number of sequences and finite number of distinct real numbers). He proved the following:

Theorem 1.1. ([16, Simplified version of Osikiewicz Theorem])

Assume that (x_n) is a splice over a partition {E_i}. Letyi=limn→∞,n∈Eixn. Assume that d(E_i) exists for each i and

Then

In fact Osikiewicz considered a more general case, namely a regular matrix summability method A and A-density the details of which are presented in the preliminaries. Very recently in [2] a new approach was made to study the general version of Osikiewicz Theorem by defining the notion of the A-density of a point and an alternative version of the same result was established. In fact it was shown that the assumptions of Osikiewicz Theorem imply those of the following Theorem:

Theorem 1.2

Suppose that x = (x_n) is a bounded sequence, δ_A(y) exists for every y ∈ ℝ and∑y∈DδA(y)=1. Then

and consequently Osikiewicz result is a consequence of Theorem 1.2.

A natural question arises whether the result is only true for regular matrices. Note that one of the main condition of regularity of a non-negative matrix A = (a_{n, k}) is that limn→∞∑k=1∞an,k=1. In this note we first show that actually the result can be extended to a larger class of matrices A = (a_{n, k}) satisfying limn→∞∑k=1∞an,k<∞ using almost the same arguments used in [2]. The main observation which leads to Theorem 2 is that if (x_n) ∈ ℓ^∞ and δ_A(y) exists for all y ∈ ℝ, then D:= {y ∈ ℝ: δ_A(y) > 0} is countable. We produce an example of a matrix A = (a_{n, k}) with limn→∞∑k=1∞an,k=∞ and a bounded sequence (x_n) for which δ_A(y) exists for all y ∈ ℝ but D:= {y ∈ ℝ: δ_A(y) > 0} is uncountable. This surprising example naturally gives rise to the following conjecture:

Conjecture (★)

For any matrix A = (a_{n, k}) with limn→∞∑k=1∞an,k=∞ there exists a bounded sequence (x_n) for which δ_A(y) exists for all y ∈ ℝ and D:= {y ∈ ℝ: δ_A(y) > 0} is uncountable.

In the last section of the note we deal with this conjecture essentially showing that it is false however showing that with imposition of certain conditions on the matrix the conjecture becomes true.

We first present the necessary definitions and notations which will form the background of this article.

If x = (x_n) is a sequence and A = (a_{n, k}) is a summability matrix, then by Ax we denote the sequence ((Ax)₁, (Ax)₂, (Ax)₃, …) where (Ax)n=∑k=1∞an,kxk. The matrix A is called regular if limn→∞xn=L implies limn→∞(Ax)n=L. The well-known Silverman-Toeplitz theorem characterizes regular matrices in the following way. A matrix A is regular if and only if

• limn→∞an,k=0,

• limn→∞∑k=1∞an,k=1,

• supn∈ℕ∑k=1∞∣an,k∣<∞.

For a non-negative regular matrix A and E ⊂ ℕ, following Freedman and Sember [9], the A-density of E, denoted by δ_A(E), is defined as follows

where 1_E is a 0–1 sequence such that 1_E(k) = 1 ⇔ k ∈ E. If δA¯(E)=δA_(E) then we say that the A-density of E exists and it is denoted by δ_A(E). Clearly, if A is the Cesàro matrix i.e.

then δ_A coincides with the natural density.

Throughout by ℓ^∞ we denote the set of all bounded sequences of reals.

In [2] in a new approach, the authors had defined for a sequence (x_n) a density δ_A(y) of indices of those x_n which are close to y which was not dealt with till then in the literature. This was a more general approach than that of Osikiewicz [16].

Fix (x_n) ∈ ℓ^∞. For y ∈ ℝ let

and

If δA¯(y)=δA_(y), then the common value is denoted by δ_A(y).

The main result of [2] was the following.

Theorem 2.1

Let x = (x_n) ∈ ℓ^∞. Suppose that the set of limit points of (x_n) is countable and δ_A(y) exists for any y ∈ ℝ where A is a non-negative regular matrix. Then

Now recall that a non-empty family ℐ of subsets of ℕ is an ideal in ℕ if for A, B ⊂ ℕ, (i) A, B ∈ ℐ ⇒ A ∪ B ∈ ℐ; (ii)A ∈ ℐ, B ⊂ A ⇒ B ∈ ℐ. Further if ∪A∈JA=ℕ ie. {n} ∈ ℐ ∀ n ∈ ℕ, then ℐ is called admissible or free. An ideal ℐ is called a P-ideal if for any sequence of sets (D_n) from ℐ, there is another sequence of sets (C_n) in ℐ such that D_n ▵ C_n is finite for all n and ∪nCn∈J. Equivalently if for each sequence (A_n) of sets from ℐ there exists A_∞ ∈ ℐ such that A_n A_∞ is finite for all n ∈ ℕ then ℐ becomes a P-ideal.

For a bounded sequence (x_n), we now recall the following definitions (see [13]):

• (x_n) is ℐ convergent to y if for any ɛ > 0, {n: |x_n − y| ≥ ɛ} ∈ ℐ.

• A point y is called an ℐ-cluster point of (x_n) if {n: |x_n − y| ≤ ɛ} ∉ ℐ for any ɛ > 0.

• y is called an ℐ-limit point of (x_n) if there is a set B ⊂ ℕ, B ∉ ℐ, such that limn∈Bxn=y.

In this note we primarily consider non-negative matrices A = (a_{n, k}) satisfying

• a_{n, k} ≥ 0 for all n, k;

• limn→∞an,k=0, for all k;

• limn∑k=1∞an,k<∞.

One should note that one can not replace the limn in (iii) above by supn for the simple fact that in that case if one considers a matrix A having infinitely many zero rows, like the matrix A = (a_{n, k}) where

then even δ_A(ℕ) does not exist.

Throughout the next section by a non-negative matrix A we will always mean a matrix satisfying the above three conditions. It should be noted that the notion A-density can be similarly defined as in the case of regular matrices and all the axiomatic conditions, as stated in [9] are satisfied here except for the fact that the density of the set of natural numbers ℕ is now a finite real number, not necessarily 1. Further it is easy to check that {E ⊂ ℕ: δ_A(E) = 0} forms a P-ideal of ℕ. The proof being very similar to the case of regular matrices (see [2]) is omitted here.

The main result which we are going to establish in this paper is the following.

Theorem 3.1

Let x = (x_n) ∈ ℓ^∞, and the set of limit points of (x_n) is countable. Let A be a non-negative matrix as defined before (necessarily withsupn∑k=1∞an,k<∞). Suppose δ_A(y) exists for all y ∈ ℝ. Then

As in [2] we start with the following observation.

Lemma 3.2

Let (x_n) ∈ ℓ^∞and δ_A(y) exists for all y ∈ ℝ. Then D:= {y ∈ ℝ: δ_A(y) > 0} is countable and

wheresupn∑k=1∞an,k=M.

Theorem 3.3

Let (x_n) be a bounded sequence and A be a non-negative matrix such that δ_A(y) exists for every y ∈ ℝ and moreover∑y∈DδA(y)=M. Then

Proposition 3.4

Suppose x = (x_n) ∈ l^∞. If δ_A(y) = M, then My is a limit point of the sequence {(Ax)_n}.

Corollary 3.5

Let (x_n) be a bounded sequence. Suppose that there are y and z (y ≠ z) with δ_A(y) = δ_A(z) = M. Then the limitlimn→∞(Ax)ndoes not exist.

Now we recall some important results from [2] which will be useful in the sequel.

Lemma 3.6.([2])

Let ℐ be an ideal of subsets of ℕ. Assume that X:= {n: x_n ∈ [a, b]} ∉ ℐ. Suppose that

for any t ∈ (a, b) and any ɛ > 0 such that ɛ < min{t − a, b − t}. Then there is y ∈ [a, b] such that {n: |x_n − y| ≥ ɛ} ∈ ℐ for every ɛ > 0.

Proposition 3.7.([2])

Let ℐ be a P-ideal. Assume that (x_n) ∈ ℓ^∞does not have any ℐ-limit points. Then the set of limit points of (x_n), i.e. the set

is uncountable and closed.

Corollary 3.8.([2])

Let [a, b] be a fixed interval and ℐ be a P-ideal. Assume that {n: x_n ∈ [a, b]} ∉ ℐ and any point y ∈ (a, b) is not an ℐ-limit point of (x_n). Then the set of limit points of (x_n) in [a, b], i.e. the set

is uncountable and closed.

Corollary 3.9.([2])

Let (x_n) ∈ ℓ^∞. Assume that the set of limit points of (x_n) is countable. Then the sequence (x_n) has at least one ℐ-limit point for every P-ideal ℐ.

We can easily prove the following results analogous to the results of [2] which will help us to reach our final goal.

Lemma 3.10

Let r ∈ (0, 1), r₁ ≥ r₂ ≥ r₃ ≥ …, limn→∞rn=rand let (E_n) be a decreasing sequence of subsets of ℕ.

• Ifδ_A(E_n) = r_n, n ∈ ℕ, then there is a subset E of ℕ withδ_A(E) = r and such that E ⊂^*E_n, n ∈ ℕ, i.e. E E_n is finite for every n ∈ ℕ. Moreover, ifδA¯(En)→r, then δ_A(E) = r.

• IfδA¯(En)=rn, n ∈ ℕ, then there is a subset E of ℕ withδA¯(E)=rand such that E ⊂^*E_n, n ∈ ℕ.

Theorem 3.11

Let (x_n) ∈ ℓ^∞. A point y ∈ ℝ is an A-statistical limit point of (x_n) if and only ifδA¯(y)>0. Moreover if δ_A(y) > 0, then there is E ⊂ ℕ with δ_A(E) = δ_A(y) andlimn∈Exn=y.

Corollary 3.12

Let (x_n) ∈ ℓ^∞. A point y ∈ ℝ is an A-statistical cluster point of (x_n) and it is not an A-statistical limit point if and only if

• δA¯({j:∣xj-y∣≤1/n})>0for every n;

• δA(y)=limn→∞δA¯({j:∣xj-y∣≤1/n})=0.

Proposition 3.13

Let (x_n) ∈ ℓ^∞. Assume that y₁, y₂, … are the only distinct real numbers such that δ_A(y_i) > 0 ∀i. Then there exists a partition E₁, E₂, … such that δ_A(E_i) = δ_A(y_i), i = 1, 2, … andlimn∈Eixn=yi.

Lemma 3.14

Assume that {E_n: n = 1, 2, … } is a partition of ℕ such that∑n=1∞δA(En)<M. Then there is a partition {F_n: n = 0, 1, 2, … } of ℕ such that

• F_n ⊂ E_n;

• δ_A(F_n) = δ_A(E_n);

• δA(F0)=M-∑n=1∞δA(En).

Finally we prove a sufficient condition for a bounded sequence (x_n) to have the property that ∑y∈ℝδA(y)=M.

Theorem 3.15

Let (x_n) be a bounded sequence. Suppose that the set of limit points of (x_n) is countable and δ_A(y) exists for all y ∈ D. Then∑y∈DδA(y)=M.

Throughout the paper we have considered the matrices with the following properties.

• a_{n, k} ≥ 0 for all n, k;

• limn→∞an,k=0, for all k;

• limn∑k=1∞an,k<∞.

Now the natural question arises whether the results discussed above remain true if any one of the conditions for the matrix can be relaxed. The first condition is clearly essential.

Coming to the second condition, if limn→∞an,k>0 for some k, then note that the axiomatic condition in [9] namely the condition ’for E₁, E₂ ⊂ ℕ such that E₁▵E₂ is finite, δ_A(E₁) = δ_A(E₂)’, which is very crucial in defining a density function does not hold anymore. For example, let

Here a_n,1 → 1 as n→∞. We take E₁ = ℕ, E₂ = ℕ {1}. Then E₁▵E₂ = {1}, but δ_A(E₁) = 2 whereas δ_A(E₂) = 1.

Finally let us relax condition (iii) and assume that limn∑k=1∞an,k=∞. Observe that Lemma 3.6 forms the backbone of our main result as only then the sequence can be partitioned into countably infinitely many splices. Below we produce an example of a matrix A with limn∑k=1∞an,k=∞ and a bounded real sequence (x_n) for which {y ∈ ℝ: δ_A(y) > 0} is uncountable, i.e. Lemma 3.2 is not true when M = ∞.

Proposition 4.1

Define the matrix A = (a_{n, k}) in the following way

where g: ℕ → (0, ∞) is defined as g(k) = n + 1 if k ∈ [2ⁿ, 2ⁿ⁺¹). Then there is a bounded sequence (x_n) such that {y ∈ ℝ: δ_A(y) > 0} is uncountable.

Conjecture (★)

For any matrix A = (a_{n, k}) with

• a_{n, k} ≥ 0 for all n, k,

• limn→∞an,k=0 for all k,

• limn→∞∑k=1∞an,k=∞.

there exists a bounded sequence (x_n) for which δ_A(y) exists for all y ∈ ℝ and D:= {y ∈ ℝ: δ_A(y) > 0} is uncountable.

Proposition 4.2

There is a non-negative matrix A = (a_{n, k}) satisfying all the above properties (as prescribed in Conjecture (★) such that for any bounded sequence (x_n) for which δ_A(y) exists for all y ∈ ℝ the set D = {y: δ_A(y) > 0} is a singleton.

Proposition 4.3

Let A = (a_{n, k}) be a non-negative matrix satisfying the conditions of Conjecture (★). In addition let there exist a sequence of positive real numbers(δk)k=1∞with the following properties.

• For any k, min{a₁₁, a₁₂, …, a_1k} ≥ δ_k;

• limn→∞δn.2n-p>0, for any p < n.