Invariant measure of rotational beta expansion and a problem of Tarski

We study invariant measures of a piecewise expanding map in $\mathbb{R}^m$ defined by an expanding similitude modulo lattice. Using the result of Bang on a problem of Tarski, we show that when the similarity ratio is not less than $m+1$, it has an absolutely continuous invariant measure equivalent to the $m$-dimensional Lebesgue measure, under some mild assumption on the fundamental domain. Applying the method to the case $m=2$, we obtain an alternative proof of the result in Akiyama-Caalim:2015 together with some improvement.