In this paper we introduce and study a certain intricate Cantor-like set ${¥mathcal C}$ contained in unit interval. Our main result is to show that the set ${¥mathcal C}$ itself, as well as the set of dissipative points within ${¥mathcal C}$, both have Hausdorff dimension equal to 1. The proof uses the transience of a certain non-symmetric Cauchy-type random walk.
