On uniform distribution of alpha beta-orbits

Let alpha, beta is an element of (0, 1) such that at least one of them is irrational. We take a random walk on the real line such that the choice of alpha and beta has equal probability 1/2. We prove that almost surely the alpha beta-orbit is uniformly distributed module one, and the exponential sums along its orbit have the square root cancellation. We also show that the exceptional set in the probability space, which does not have the property of uniform distribution modulo one, is large in the terms of topology and Hausdorff dimension. (c) 2020 Elsevier Inc. All rights reserved.