A formula for calculation of metric dimension of converging sequences (Q2761039)

Let \(X\) be a totally bounded metric space and let \(\underline {\dim}X\) \((\overline {\dim} X)\) be a lower (upper) metric dimension of \(X\), respectively. NEWLINENEWLINENEWLINEThe aim of this paper is to give analytic formulae for the calculations of metric dimensions mentioned above in a particular case of countable \(X\). Namely, the authors consider here only \(c\)-sequences -- the class of decreasing sequences of positive numbers converging to zero \(\{a_n\}^\infty_{n=1}\), which are given by (i) decreasing, (ii) differentiable, and (iii) convex functions \(f\) with (iv) limit zero at infinity i.e. \(a_n=f(n)\), \(n\in \mathbb N\), for some function \(f\) with properties (i)--(iv). For any \(c\)-sequence \(\{a_n\}^\infty_{n=1}\) we put now \(X=A=\bigcup_{n\in \mathbb N}\{a_n\}\cup \{0\}\). NEWLINENEWLINENEWLINEThus, we can present the main result of the paper (cf. theorem on pp. 397-398) as follows: NEWLINENEWLINENEWLINESuppose that \(c\)-sequence \(\{a_n\}^\infty_{n=1}\) is given by \(f\). Then NEWLINE\[NEWLINE \underline {\text{dim}} A = \liminf_{x\rightarrow \infty} \frac {\log \big (x-\frac {f(x)}{f'(x)}\big)}{-\log (-f'(x))},\quad \overline {\text{dim}}A=\limsup_{x\rightarrow \infty} \frac {\log \big (x -\frac {f(x)}{f'(x)}\big)}{-\log (-f'(x))}. NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEIn conclusion, some corollaries and numerical examples are added.