Mixing automorphisms of compact groups and a theorem by Kurt Mahler (Q1123990)

Let X be a compact metrizable abelian group and let N be its (discrete) dual group. Suppose there is an action \(\alpha\) of \({\mathbb{Z}}^ d\) on X by automorphisms \(\alpha_{\underset \tilde{} n}\) (\(\underset \tilde{} n\in {\mathbb{Z}}^ d)\). Let \(R_ d={\mathbb{Z}}[u_ 1^{\pm 1},...,u_ d^{\pm 1}]\) denote the ring of Laurent polynomials with integer coefficients in d variables. In a natural way, N is in \(R_ d\) module; if \(\underset \tilde{} n=(n_ 1,...,n_ d)\), the automorphism \(\alpha_{\underset \tilde{} n}\) on X corresponds to \(a\mapsto u_ 1^{n_ 1}...u_ d^{n_ d}.a\) on N. This paper is one of a series in which dynamical properties of the action are discussed using algebraic properties of the module. (The reader should look out for the author's book ``Algebraic ideas in ergodic theory'' which will appear in the CBMS series of the American Mathematical Society.) One particular feature of the correspondence between dynamics and algebra is that (X,\(\alpha)\) satisfies the descending chain condition (that is, every decreasing sequence of \(\alpha\)-invariant closed subgroups is eventually constant) if and only if N is finitely generated. The action \(\alpha\) is r-mixing if for all measurable \(B_ 0,B_ 1,...,B_ r\) in X \[ \lim \lambda (B_ 0\cap \alpha_{\underset \tilde{} n_ 1}(B_ 1)\cap...\cap \alpha_{\underset \tilde{} n_ r}(B_ r))=\lambda (B_ 0).\lambda (B_ 1)...\lambda (B_ r), \] where \(\lambda\) is Haar measure and the limit is taken over all sets \(\{\) \(\underset \tilde{} n_ 1,...,\underset \tilde{} n_ r\}\subseteq {\mathbb{Z}}^ d\) with \(n_ i\to \infty\) and \(n_ i-n_ j\to \infty\) if \(i\neq j\). Mixing means 1-mixing. A shape is a subset \(\{\) \(\underset \tilde{} n_ 0,\underset \tilde{} n_ 1,...,\underset \tilde{} n_ r\}\) of \({\mathbb{Z}}^ d\); the shape is mixing if (with \(B_ 0,...,B_ r\) as above) \[ \lim_{k\to \infty}\lambda (\alpha_{k\underset \tilde{} n_ 0}(B_ 0)\cap...\cap \alpha_{k\underset \tilde{} n_ r}(B_ r))=\lambda (B_ 0)...\lambda (B_ r). \] Thus if \(\alpha\) is r-mixing, every shape with \(r+1\) elements is mixing. The main result of the paper (Theorem 3.8) considers a mixing \(\alpha\) for which (X,\(\alpha)\) satisfies the descending chain condition. (1) If X is connected, every shape is mixing. (2) If X is zero-dimensional, then every shape is mixing if and only if the set \(\{p_ 1,...,p_ m\}\) of the prime ideals of \(R_ d\) which are the annihilators of elements of N satisfies \(p_ i=\pi_ iR_ d\), where \(\pi_ 1,...,\pi_ m\) are distinct rational primes in \({\mathbb{Z}}.\) The proof of (1) above uses the following extension of a theorem of Mahler. Let K be a field of characteristic 0. Let \(\alpha_ 1,...,\alpha_ q\) be q (\(\geq 2)\) units in K. Suppose there exist units \(c_ 1,...,c_ q\) in K such that the equality \(c_ 1\alpha^ n_ 1+...+c_ q\alpha^ n_ q=0\) holds for infinitely many values of \(n\in {\mathbb{N}}\); then it holds for n belonging to some non-trivial arithmetic progression in \({\mathbb{N}}\).