Let $A \subseteq \mathbb{N}$, define the upper density of $A$ as, $$ \overline{\delta}(A) := \limsup_{N\to\infty}\frac{|A\cap\{1,2,3,\cdots,N\}|}{N}. $$ This naturally leads to a weaker form of convergence of a sequence $(x_n)$ in $\mathbb{R}$:

A sequence $(x_n)$ converges in density to $x\in \mathbb{R}$ if for every $\epsilon>0$, the set, $$ \{n\in \mathbb{N}: |x_n - x| \geq \epsilon\} $$ has upper density zero. Denote this as $D\!-\! \lim_{n\to\infty}x_n =x$. This form of convergence is important in applications of Ergodic Theory to Ramsey Theory.

One can then consider the closed subspace of $\ell^\infty$, $$ \overline{\delta}c_0 := \{(x_n)\in \ell^\infty:D\!-\! \lim_{n\to\infty}x_n = 0\} $$ I would like to know if there are any references where the $\overline{\delta}c_0$ is studied in $\ell^\infty$.

Thanks in advance!


1 Answer 1


This type of convergence is often called statistical convergence.

The paper Constantin P. Niculescu, Gabriel T. Prajitura: Some open problems concerning the convergence of positive series (arXiv:1201.5156) mentioned in connection with the history of this notion that: "The monograph of H. Furstenberg [13] outlines the importance of convergence in density in ergodic theory. In connection to series summation, the concept of convergence in density was rediscovered (under the name of statistical convergence) by Steinhaus [28] and Fast [12] (who mentioned also the first edition of Zygmund’s monograph [31], published in Warsaw in 1935)."

Statistical convergence can be viewed as a special case of ideal convergence for the ideal of density zero sets $$\mathcal I=\{A\subseteq\mathbb N; \delta(A)=0\}.$$ (Or, if you prefer filters, you can take filter convergence w.r.t. the filter of sets with the full density $\delta(A)=1$.)

Of course, since the function $x\mapsto\operatorname{\mathcal I-lim} x_n$ is continuous w.r.t. the sup-norm, the set $c_0(\mathcal I)$ of all bounded $\mathcal I$-null sequences is a closed linear subspace of $\ell_\infty$.

If you are interested also in this generalization, i.e., the space of $$c_0(\mathcal I)=\{x\in\ell_\infty; \operatorname{\mathcal I-lim} x_n=0\},$$ then some related papers could be, for example:

  • $\begingroup$ This is not exactly what you asked about - the space you are interested in is a special case for the ideal of sets with zero density. But I hope that at least the information that there are some papers which study this generalization might be helpful - and possibly it might help you to find further resources. $\endgroup$ Commented Oct 30, 2018 at 12:16
  • $\begingroup$ Thank you very much. This is exactly what I was looking for. I wasn't aware of the term 'statistical convergence'. $\endgroup$ Commented Oct 30, 2018 at 14:55
  • $\begingroup$ @MartinSleziak Thanks for the nice answer, I didn't know the article of T. Šalát, B.C. Tripathy, M. Ziman. As a side remark, I think it would be more natural to define $c_0(\mathcal{I})$ as the set of sequences (not necessarily bounded) which are $\mathcal{I}$-convergent to $0$. $\endgroup$ Commented Dec 21, 2018 at 11:08

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