A Note on Fractal Measures and Cartesian Product Sets
| Autor: | Meriem Ben Hadj Khlifa, Najmeddine Attia, Hajer Jebali |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Bulletin of the Malaysian Mathematical Sciences Society. 44:4383-4404 |
| ISSN: | 2180-4206 0126-6705 |
| DOI: | 10.1007/s40840-021-01172-1 |
| Popis: | In this paper, we give a new product formula : $$\begin{aligned} {\mathsf {H}}^t(E) {\mathsf {H}}^s(F)\le c_1 {\mathsf {H}}^{t+s}(E\times F)\le & {} c_2 {\mathsf {H}}^s(E) {\mathsf {P}}^t(F)\\\le & {} c_3 {\mathsf {P}}^{t+s}(E\times F) \le c_4 {\mathsf {P}}^s(E) {\mathsf {P}}^t(F). \end{aligned}$$ where $$E\subseteq \mathbb {R}^d$$ , $$F\subseteq \mathbb {R}^l$$ , $$t,s\ge 0$$ and $$ {\mathsf {H}}^t$$ and $${\mathsf {P}}^s$$ denote, respectively, the lower and upper Hewitt–Stromberg measures. Using these inequalities, we give lower and upper bounds for the lower and upper Hewitt–Stromberg dimensions $${\mathsf {b}}(E\times F)$$ and $${\mathsf {B}}(E\times F)$$ in terms of the Hewitt–Strombeg dimensions of E and F. |
| Databáze: | OpenAIRE |
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