We give a simple proof of the theorem by Dembo, Peres, Rosen and Zeitouni (DPRZ) regarding the time Brownian motion needs to cover every $\varepsilon$ ball on the two-dimensional unit torus in the $\varepsilon\searrow 0$ limit.
"A simple proof of the DPRZ theorem for 2d cover times." Ann. Probab. 48 (1) 445 - 457, January 2020. https://doi.org/10.1214/19-AOP1366