Uniform density u and corresponding I u - convergence

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1 Mathematical Communications (2006), -7 Uniform density u and corresponding I u - convergence Vladimir Baláž and Tibor Šalát Abstract. The concept of a uniform density of subsets A of the set N of positive integers was introduced in [] and [2]. Corresponding I u - convergence to the notion of uniform density u can be found in [8]. This paper studies I u - convergence in detail. Key words: uniform density, I u - convergence, almost convergence, strong p -Cesàro convergence AMS subject classifications: Primary 40A05, 40D25; Secondary C05 Received May 7, 2005 Accepted January 27, 2006 We recall some nown notions. Let A N. Ifm, n N, bya(m, n) wedenote the cardinality of the set A [m, n]. Numbers A(,n) d(a) = lim inf, d(a) = lim supa(,n) are called the lower and the upper asymptotic density of the set A, respectively. If there exists the limit lim supa(,n), then d(a) = d(a) = d(a) is said to be the asymptotic density of A. The uniform density of A N was introduced in [] and [2] as follows: Put a n =mina(m +,m+ n), an =maxa(m +,m+ n). It can be shown (see [2]) that the following limits exist a n u(a) = lim, a n u(a) = lim The wor on this paper was supported by GRANT VEGA /2005/05 and GRANT VEGA /0286/03 Department of Mathematics, Chemical Technology Faculty, Slova Technical University, Radlinsého 9, SK-8237 Bratislava, Slovaia, vladimir.balaz@stuba.s Mathematico-fyziálna faulta, Universita Komensého, Mlýnsa dolina, SK-8425, Slovaia

2 2 V. Baláž and T. Šalát and they are called the lower and the upper uniform density of the set A, respectively. If u(a) = u(a), then u(a) = u(a) is called the uniform density of A. It is clear that for each A N we have u(a) d(a) d(a) u(a). () Hence if there exists u(a), then there also exists d(a) andu(a) = d(a). The converse is not true (see Example.). The concept of statistical convergence was introduced in [4] (see also [3], [5], [0], []) as follows: Let x =(x n ) be a sequence of complex numbers. The sequence x is said to be statistically convergent to a complex number L provided that for every ɛ>0wehaved(a ɛ )=0,whereA ɛ = {n N : x n L ɛ}. Ifx =(x n ) converges statistically to L, then we write lim - stat x n = L. A generalized approach to convergence is done in [6] by means of the notion of an ideal I of subsets of N (i.e. I is an additive and hereditary class of sets). A sequence x is said to be I -convergenttol provided that for every ɛ>0the set A ɛ belongs to I, wewritei - lim x n = L. PutI = I d = {A N : d(a) = 0}, then I d - convergence coincides with statistical convergence. Hence lim - stat x n = L = I d - lim x n.inthecasei = I u = {A N : u(a) =0} we obtain I u - convergence. If x =(x n ) is I u -convergenttol, wewritei u - lim x n = L. We can easily verify that if I u - lim x n = L, I u - lim y n = L 2,thenI u - lim (x n + y n )=L + L 2 and if a is constant, then I u - lim ax n = al. By M we denote the set of all I u - convergent sequences; M is a linear space. Analogously, we have for M 0, the set of all statistically convergent sequences (see []). Let c be the set of all convergent sequences. By () we have c M M 0. The following examples show that c M and M M 0 even in case of bounded sequences. Example. Let P be the set of all primes. Define x =for P and x =0 otherwise. Because of u(p )=0(see [2]), we have that x =(x ) is I u - convergent to 0, but not convergent. Example 2. It is easy to see that for the set A = {0 +, 0 +2,...,0 + } = we have d(a) =0, u(a) =0, u(a) =. Put x =for A and x =0for / A. ThenI d -limx =0,butx =(x ) is not I u - convergent. We recall the notion of strong p -Cesàro convergence and almost convergence. A sequence x =(x ) is said to be strong p -Cesàro convergent (0 <p< ) toa number L if n lim x L p =0 = (see [3]). By w p denote the set of all strong p -Cesàro convergent sequences. A bounded sequence x =(x ) is almost convergent to a number L if every Banach limit of x is equal to L, which is equivalent to the condition p lim x n+i = L p p i=

3 Uniform density u and corresponding I u - convergence 3 uniformly in n(see [9], [0], p 59-62). By F we denote the set of all almost convergent sequences. It is shown in [9] that almost convergence and statistical convergence are not compatible even in the case of bounded sequences. The following Theorem shows that in the case of bounded sequences I u -convergence and almost convergence can be compared. Theorem. Suppose x =(x ) is a bounded sequence. If x is I u - convergent to L, thenx is almost convergent to L. Proof. Let p, n N be arbitrary. We estimate S(n, p) = x n+ + x n x n+p L. p We have S(n, p) S () (n, p)+s (2) (n, p), (2) where S () (n, p) = p S (2) (n, p) = p j p, n + j A ɛ j p, n + j / A ɛ x n+j L, x n+j L. By using the definition of A ɛ = {n N : x n L ɛ} we have S (2) (n, p) <ɛ for every n =, 2,.... (3) The boundedness of x =(x ) implies that there exists M>0 such that x L M ( =, 2,...). (4) Then (4) implies S () (n, p) M A ɛ(n +,n+ p) p max A ɛ(m +,n+ p) m M = M ap p p. Using the last estimation which holds for every n =, 2,... and (2), (3) we obtain the assertion of Theorem. Remar. The converse of the previous theorem does not hold. For instance, let y =(y ) be the sequence defined by y =if n is even and y =0if n is odd. The sequence y is almost convergent to /2 but it is not I u - convergent. In [3] a connection between strong p -Cesàro convergence and statistical convergence is articulated. In the case of bounded sequences both of these inds of convergence are equivalent. A similar result can be obtained for I u - convergence. First of all, we define a new ind of convergence, so-called uniformly strong p -Cesàro

4 4 V. Baláž and T. Šalát convergence, which is a generalization of the notion of strong almost convergence (see [8]). Definition. The sequence x =(x ) is said to be uniformly strong p-cesàro convergent (0 <p< ) toanumberl if lim n+ i=n+ x i L p =0 uniformly in n. By uw p denote the set of all uniformly strong p -Cesàro convergent sequences. It is immediate that uw p w p (0 <p< ). Example 2 shows that the inclusion is strict. Theorem 2. a) If 0 <p< and a sequence x =(x ) is uniformly strong p -Cesàro convergent to L, thenitisi u - convergent to L. b) If x =(x ) is bounded and I u - convergent to L, then it is uniformly strong p -Cesàro convergent to L for every p, 0 <p<. Proof. a) Let x be uniformly strong p -Cesàro convergent to L, 0 < p <. Suppose ɛ>0. Then, for every n N we have x n+j L p x n+j L p ɛ p A ɛ (n +,n+ ), j= and further, j= j, x n+j L ɛ x n+j L p ɛ p max A ɛ(m +,m+ ) = ɛ p a a for every n =, 2, 3....This implies lim =0,andu(A ɛ) = 0, so that I u -lim x n = L. b) Now, suppose that x is a bounded sequence and I u - lim x n = L. Let 0 <p< and ɛ>0. According to the assumption, we have u(a ɛ )=0. (5) The boundedness of x = (x ) implies that there exists M > 0 such that x L M ( =, 2,...). Observe that for every n N, wehavethat x n+j L p = j= j, n + j A ɛ x n+j L p + j, n + j / A ɛ x n+j L p max A ɛ(m +,m+ ) M + ɛ p ɛ p + M a (6)

5 Uniform density u and corresponding I u - convergence 5 Using (5) and (6) we obtain lim x n+j L p = 0, uniformly in n. j= Corollary. If x =(x ) is a bounded sequence, then x is I u - convergent to L if and only if x is uniformly strong p -Cesàro convergent to L for every p, 0 <p<. In [3], [5] and [] it is shown that statistical convergence can be characterized by the convergence in the usual sense along a great set of indexes, great in the sense of asymptotic density. The following theorem shows that the I u - convergence can be characterized by the convergence along a great set of indexes, great now being in the sense of uniform density. In [6] it is shown that a similar statement is not true for the I - convergence where I is an arbitrary ideal. Theorem 3. Asequencex =(x ) is I u - convergent to L if and only if there exists a set K = { < 2 <...< n <...} N such that u(k) =and lim x n = L. n Proof. If there exists a set with the mentioned properties and ɛ is an arbitrary positive number, we can choose a number m N such that for each n>mwe have x n L <ɛ. (7) Let A ɛ = {n N : x n L ɛ}. Then, on the basis of (7), we have A ɛ N { m+, m+2,...} where on the right-hand side there is a set with the uniform density 0. Therefore, u(a ɛ ) = 0; hence I u - lim x = L. Now suppose that a sequence x =(x ) is I u -convergenttol. LetK j be the complement of the set A /j for j =, 2,..., { K j = N n N : x n L } j Then, by the definition of I u - convergence, we have By the definition of K j we have u(k j )= for j =, 2,.... K K 2... K j K j+.... (8) Let us choose an arbitrary number s K. By the definition of K j there exists anumbers 2 >s, s 2 K 2 such that for each n s 2 we have min K 2(m +,m+ n) > n 2. Again on the basis of the definition of K j there exists a number s 3 >s 2, s 3 K 3, such that for each n s 3 we have

6 6 V. Baláž and T. Šalát min K 3(m +,m+ n) > 2 n 3. In this manner we can construct an increasing sequence of positive integers s <s 2 <...<s j <... such that s j K j and that for each n s j we have min j(m +,m+ n) > n j for j =, 2,.... (9) Define K as follows: if s,then K; suppose that j andthats j < s j+,then K if and only if K j.letk = { < 2 <... < n <...}. According to (8) and (9), for each n, s j n<s j+ we have min K(m +,m+ n) min K j(m +,m+ n) > n n j. From this it is obvious that u(k) =. Let ɛ>0 be given and select j such that /j < ɛ. Let n s j, n K. Then there exists a number r j such that s r n<s r+. According to the definition of K, n K r,wehave x n L < r j <ɛ. Thus x n L <ɛfor each n s j, n K. Hence lim x n = L. n Corollary 2. If a sequence x =(x ) is uniformly strong p -Cesàro convergent (0 <p< ) or I u - convergent to L, then there exists a sequence y =(y ) convergent to L and a sequence z =(z ) I u - convergent to 0, such that x = y + z and u(b) =0,whereB = {n N : z 0}. Proof. First observe that if x is uniformly strong p -Cesàro convergent to L (0 <p< ), then x is I u -convergenttol. From the previous theorem there exists a set K = { < 2 <...< n <...} N such that u(k) = and lim x = L. We define y and z as follows: If K, put y = z and z =0,andif/ K, we put y = L and z = x L. Remar 2. If a sequence x =(x ) is uniformly strong p -Cesàro convergent (0 <p< ) or I u - convergent to L, thenx has a subsequence which converges to L. References [] T. C. Brown, A. R. Freedman, Arithmetic progressions in lacunary sets, Mountain J. Math. 7(987),

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