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metric dimension; converging sequences; summability of series
In the paper, some kind of independence between upper metric dimension and natural order of converging sequences is shown --- for any sequence converging to zero there is a greater sequence with an arbitrary ($\leqslant 1$) upper dimension. On the other hand there is a relationship to summability of series --- the set of elements of any positive summable series must have metric dimension less than or equal to $1/2$.
[K-T] Kolmogorov A. N., Tihomirov V. M.: $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional spaces (in Russian). Usp. Mat. Nauk 14 (1959), 3-86 Am. Math. Soc. Transl. 17 (1961), 277-364. MR 0112032
[M-Z] Mišík L., Žáčik T.: On some properties of the metric dimension. Comment. Math. Univ. Carolinae 31 (1990), 781-791. MR 1091376
[P-S] Pontryagin L.S., Snirelman L.G.: Sur une propriété metrique de la dimension. Annals of Math. 33 (1932), 156-162 Appendix to the Russian translation of ``Dimension Theory'' by W. Hurewitcz and H. Wallman, Izdat. Inostr. Lit. Moscow, 1948. MR 1503042
[V] Vosburg A.C.: On the relationship between Hausdorff dimension and metric dimension. Pacific J. Math. 23 (1967), 183-187. MR 0217776 | Zbl 0153.24701
[Y] Yomdin Y.: The geometry of critical and near-critical values of differentiable mappings. Math. Ann. 264 (1983), 495-515. MR 0716263 | Zbl 0507.57019
[Z] Žáčik T.: On some approximation properties of the metric dimension. Math. Slovaca 42 (1992), 331-338. MR 1182963
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