Density-$c_0$ in $\ell^\infty$

Question

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!

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)."

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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:

• Paolo Leonetti: Continuous Projections onto Ideal Convergent Sequences, Results in Mathematics (2018), 73:114; doi: 10.1007/s00025-018-0876-8 (see also follow-up by paper by Tomasz Kania, A letter concerning Leonetti's paper `Continuous Projections onto Ideal Convergent Sequences', arXiv:1810.09383)
• T. Šalát, B.C. Tripathy, M. Ziman: On $\mathcal I$-convergence field, Italian J. of Pure and Appl. Math, 17:45-54, 2005.
• Remarks on ideal boundedness, convergence and variation of sequences, J. Math. Anal. Appl. 375 (2011) 431-435; doi: 10.1016/j.jmaa.2010.09.023, author's homepage (Here it is studied in $\ell_\infty(\mathcal I)$ rather than in $\ell_\infty$.)