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 $\theta\in [0,2\pi)$.