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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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