Abstract: In this talk, we discuss various critical random fractals whose Hausdorff dimension is 0: the pioneer triple points (PTP) of 2D Brownian motion, the pioneer double cut points (PDCP) of 2D and 3D Brownian motions, and the boundary double points (BDP) of clusters of 2D Brownian loop soup at critical intensity. We develop a unified approach to establish the non-existence of all these objects above, which answers open questions from Burdzy and Werner in 1996 (for PTP) and Qian and Werner in 2019 (for BDP). This talk is based on a joint work in progress with Yifan Gao (CityU HK), Runsheng Liu (PKU) and Wei Qian (CityU HK).