A random walk proof of the Erdős-Taylor conjecture

Our approach in that paper was to first prove an analogous result for planar Brownian motion and then to use strong approximation. The goal of this paper, which is purely expository, is to show how to prove (1.1) using only random walk methods. We also go beyond (1.1) and study the size of the set of ‘frequent points’, i.e. those points in Z2 which are visited an unusually large number of times, of order (log n)2. Perhaps more important than our specific results, we develop powerful estimates for the simple random walk which we expect will have wide applicability. Let Ln denote the number of times that x ∈ Z2 is visited by the random walk in Z2 up to step n, (so that Ln = maxx∈Z2 L x n). Set