Uniform distribution in the $(3x+1)$-problem

Structure theorem of the $(3x+1)$-problem claims that the images under $T^n$ of arithmetic progressions with step $2^k$ are arithmetic progressions with step $3^m$. Here $T$ is the basic underlying map and a given $3^m$ progression can be the image of many different $2^k$ progressions. This gives rise to a probability distribution on the space of $3^m$ progressions. In this paper it is shown that this distribution is in a sense close to the uniform law.