The visits to zero of a random walk driven by an irrational rotation

We give a detailed analysis of the returns to zero of the “deterministic random walk” (Formula presented.) where α is a quadratic irrational, (Formula presented.), and x is sampled uniformly in [0, 1]. The method is to find the asymptotic behavior of the ergodic sums of L¹ functions for linear flows on the infinite staircase surface. Our methods also provide a new proof of J. Beck’s central limit theorem for S_n(0) where n ∈ {1, …,N} is uniform and N → ∞, and they allow us to determine the generic points for certain infinite measure preserving skew products (“cylinder maps”).