Let Tn(x) denote the time of first visit of a point x on the lattice torus Zn = Z /nZ by the simple random walk. The size of the set of α, n-late points Ln(α) = {x ∈ Zn :Tn(x) ≥ α 4 π (n logn) } is approximately n, for α ∈ (0,1) [Ln(α) is empty if α> 1 and n is large enough]. These sets have interesting clustering and fractal properties: we show that for… (More)