Projecting the one-dimensional Sierpiński gasket
The paper under review deals with the projections of the Sierpiński gasket. One considers the linear projection \(\pi_u\) of \({\mathbb{R}}^2\) onto the \(x\)-axis that maps \((0,1)\) to the point \((u,0)\). Denote \(S_u=\pi_u(S)\). The author computes the Lebesgue measure and the Hausdorff dimension of the projection \(S_u\). He proves that the projection of \(S\) in any irrational direction has Lebesgue measure equal to zero, while the projection in a rational direction \(p/q\) has Hausdorff dimension less than 1 unless \(p+q=0 \text{mod} 3\), in which case the projection has Lebesgue measure equal to \(1/q\). A lower bound for the Hausdorff dimension of \(S_u\) is also given in terms of the approximants of \(u\) by rationals. Moreover, it is constructed a residual set of directions on which the Hausdorff dimension of \(S_u\) is 1.
scientific article; zbMATH DE number 1011400