Ministry and Higher Education grant no. [DI2012 002942]{}; Piotr Oprocha was supported by the Polish Ministry of Science and Higher Education from sources for science in the years 2013-2014, grant no. IP2012 004272.

The idea of writing this paper together was born in June 2014 during the Activity on Dynamics & Numbers in the Max Planck Institute of Mathematics in Bonn. The hospitality of MPI is gratefully acknowledged. A preliminary version was derived from the set of notes prepared by the first named author for the mini-course *Specification and shadowing properties* presented by him during *A week on Dynamical Systems* at IM-UFRJ in Rio de Janeiro. Using this occasion he would like to thank the organizers for the invitation and warm reception.

Basic definitions and notation
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Notation and some conventions
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We write ${\mathbb{N}}=\{1,2,3,\ldots\}$ and ${{{\mathbb{N}}_0}}=\{0,1,2,\ldots\}$. By $|A|$ we mean the cardinality of a finite set $A$. Given any set $A\subset{{{\mathbb{N}}_0}}$ we write

-   $\overline d(A)$ for the *upper asymptotic density* of $A$, that is, $$\overline d(A)=\limsup\limits_{n\to\infty}\frac{\big|A\cap\{0,\ldots,n-1\}\big|}{n},$$

-   $\operatorname{BD^*}(A)$ for the *upper Banach density* of $A$, that is, $$\operatorname{BD^*}(A)=\limsup\limits_{n\to\infty}\max\limits_{k\in{{{\mathbb{N}}_0}}}\frac{\big|A\cap\{k,k+1,\ldots, k+n-1\}\big|}{n}.$$

We denote the set of all sequences ${\underline{x}=\{x_n\}_{n=0}^{\infty}}$ with $x_n$ in some (not necessarily finite) set $A$ for $n=0,1,\ldots$ 