Invariant Measure of Rotational Beta Expansion and Tarski’s Plank Problem

Abstract

We study invariant measures of a piecewise expanding map in R m defined by an expanding similitude modulo lattice. Using the result of Bang [5] on the plank 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 mdimensional 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 [1] together with some improvement.

Representations for complex numbers with integer digits

“…Observe that our assumption on ε implies that 0 ≤ ε |ζ − 1| 2 < 1, hence our choice of μ 2 is meaningful and z ∈ D. Observing (3.4) yields 2 and observe that the assumption on ε implies δ to be strictly positive. Since Re (ζ ) > 0 we can choose…”

“…For characterising pairs that have the finiteness property ( F ) we can use so-called shift radix systems. By shift radix systems we mean a family of -actions that are related with Canonical number systems as well as beta-expansions (see [ 2 ]). We show that there is an analogue connection with the zeta-transformation.…”

“…Note that shift radix systems have been originally defined in [ 2 ] for only. Later, the case has been introduced in [ 6 ] as symmetric shift radix systems.…”

“…The corollary is a complex version of [ 2 , Theorem 2.1], [ 6 , Theorem 3.7] and [ 56 , Theorem 3.4].…”

“…The case ε = 1 /2 on the right is the most universal case, that is the bases in the grey area do not provide unique zeta-expansions for any choice of ε Finally we are concerned with bases ζ that are algebraic integers. We will see that the behaviour of the zeta-transformation S on the elements of Z[ζ ] can be described with the aid of shift radix systems (see [2]). In fact, our result is quite analogous to the well-known relation between shift radix systems and beta-expansions with respect to algebraic integer.…”