A generalization of a result of R. Lyons about measures on \([0,1)\) (Q1261893)

The following is proved: Let \(\mu\) be an invariant measure on \([0,1]\) for \(S(x)= px\text{ mod }1\), where \(p\geq 1\) is an integer. Suppose that there is no invariant component of zero entropy. Denote \(T_ v(x)= vx\) for \(v\geq 1\). Then, for \(\varepsilon>0\), there exist \(b\), \(v\), and \(m\) with \(p^ m b\leq v\leq p^ m(b+1)\) such that the measure \(\mu\circ T^{- 1}_ v\) is \(\varepsilon\)-close to Lebesgue measure (in a metric for the weak topology). The theorem contains results by Furstenberg, Lyons, Rudolph and Johnson.