On a Construction of Plane Sets with Pre-assigned Metric Dimensions

Abstract

In this paper, some results of S.-M. Jung[3] will be generalized as follows: For a given 0<r<2, a compact and uncountable plane set Cr which satisfies dim_B C_r=r and ,for all ,shall be constructed. This gives negative answers to the question whether Marstrand's theorem is also true when 'Hausdorff dimension's in the statement of the theorem are replaced by 'metric dimension's. In the case when 0<s<1, we shall investigate a method to construct compact uncountable plane sets E with .Furthermore, we shall construct a compact and uncountable plane set F with dim_B F>1 and dim_B proj F=1 for all .