Metric discrepancy results for geometric progressions perturbed by irrational rotations

For $$\theta \in (-\infty , -1)\cup (1, \infty )$$ θ ∈ ( - ∞ , - 1 ) ∪ ( 1 , ∞ ) and for almost every x , it is known that the sequence $$\{\theta ^k x\}$$ { θ k x } is uniformly distributed modulo 1. The speed of convergence sensitively depends on the algebraic nature of $${\theta}$$ θ . In this paper we prove that such dependence vanishes if we perturb the sequence by adding the irrational rotation $$\{\kappa\gamma\}$$ { κ γ } . The speed becomes identical with that of the sequence of uniformly distributed independent random variables.