Counterexamples for metric dimensions of plane sets and their projections
Bull. Korean Math. Soc. 1996 Vol. 33, No. 1, 65-73
Soon Mo Jung
Hong-Ik University
Abstract : We ask if Marstrand's theorem is also true when the Hausdorff dimensions in the statement of the theorem are replaced by the metric dimensions. We shall give a negative answer by constructing two compact uncountable subsets $C_{1}$ and $C_{2}$ of $\R^{2}$. The set $C_{1}$ satisfies $\text{dim}_{B}C_{1}=1$ and $\overline{\text{dim}}_{B}\text{proj}_{\theta}C_{1} < 1$ for all $\theta\in [0,2\pi)$. The other set $C_{2}$ satisfies $\text{dim}_{B}C_{2} > 1$ and $\overline{\text{dim}}_{B}\text{proj}_{\theta}C_{2} < 1$ for all $\theta\in [0,2\pi)$.
Keywords : Hausdorff dimensions, metric dimensions, projection
MSC numbers : Primary 28A78; Secondary 28A80
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