On the Hausdorff dimension of general Cantor sets

Abstract

Introduction and notation. In this paper a generalization of the Cantor set is discussed. Upper and lower estimates of the Hausdorff dimension of such a set are obtained and, in particular, it is shown that the Hausdorff dimension is always positive and less than that of the underlying space. The concept of local dimension at a point is introduced and studied as a function of that point.