The Hausdorff dimension of the set of dissipative points for a   Cantor-like model set for singly cusped parabolic dynamics
  In this paper we introduce and study a certain intricate Cantor-like set $C$ contained in unit interval. Our main result is to show that the set $C$ itself, as well as the set of dissipative points within $C$, both have Hausdorff dimension equal to 1. The proof uses the transience of a certain non-symmetric Cauchy-type random walk. 
J. Schmeling, B.O. Stratmann