Hausdorff dimension of certain sets related to random αβ-orbits which are not dense

Abstract

Let α,β∈[0,1)=R/Z such that at least one of them is irrational. It is known that the random αβ-orbits of the Bernoulli mixture of rotations of α and β by choosing them with equal probability 12 are uniformly distributed modulo 1 with probability one. Chen et al. (2021) [2] showed that the exceptional set in the probability space has full Hausdorff dimension. In this note, we prove that certain sets related to random αβ-orbits which are not dense in R/Z have Hausdorff dimension 1. Our result can be viewed as an improvement of the dimension result obtained by Chen, Wang and Wen.