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Mixing automorphisms of compact groups and a theorem of Schlickewei. (English) Zbl 0824.28012
The authors show that if $$d>1$$ and $$\alpha$$ is a mixing $$\mathbb Z^ d$$- action by automorphisms of a connected compact Abelian group $$X$$, then $$\alpha$$ is mixing of all orders. In particular this is true for commuting toral automorphisms. The result is known to be false if $$X$$ is disconnected. The decidedly nontrivial proof begins by establishing an algebraic characterization of $$r$$-mixing in terms of certain prime ideals in the ring of Laurent polynomials $$\mathbb Z[u^{\pm 1}_ 1,\dots, u^{\pm 1}_ d]$$. Then they apply a deep theorem of H. P. Schlickewei [Invent. Math. 102, No. 1, 95–107 (1990; Zbl 0711.11017)] which gives a bound on the number of solutions in $$S$$-units of an algebraic number field of the equation $$a_ 1 v_ 1+\cdots+ a_ n v_ n= 1$$ for which no proper subsum vanishes.

##### MSC:
 28D15 General groups of measure-preserving transformations 22D40 Ergodic theory on groups 37A25 Ergodicity, mixing, rates of mixing 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
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##### References:
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