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Additive relations in fields: an entropy approach. (English) Zbl 0841.11035
W. M. Schmidt and H. P. Schlickewei have developed powerful subspace theorems about diophantine approximation by simultaneous linear forms with applications to $${\mathcal S}$$-units and additive relations in fields. A particular case of the latter is the following result of A. J. van der Poorten and H. P. Schlickewei [J. Aust. Math. Soc., Ser. A 51, 154–170 (1991; Zbl 0747.11017)]: If $$c_1, \dots, c_n$$ are non-zero elements and $$\Gamma$$ is a finitely generated multiplicative subgroup in a field $$\mathbb F$$ of characteristic zero, then the equation $$c_1 \gamma_1+ \cdots+ c_n \gamma_n=1$$ has only finitely many solutions in elements $$\gamma_i$$ in $$\Gamma$$ with the additional property that no proper sub-sum vanishes. Another application by the author and K. Schmidt is that a mixing action of $$\mathbb Z^d$$ by automorphisms of a compact connected abelian group is mixing of all orders.
The author seeks completely ergodic-theoretic proofs for results on additive relations and obtains this for $$\mathbb Z^d$$ actions with completely positive entropy on infinite-dimensional compact groups (these are known to be mixing of all orders). This leads to the van der Poorten-Schlickewei theorem in case the $$r$$, say, generators of $$\Gamma$$ produce an extension of the rationals of transcendence degree $$r-1$$. In this case the associated dynamical system is a finite entropy system. If the transcendence degree is less than $$r-1$$, the corresponding dynamical system has zero entropy and is still mixing of all orders, but the proof of this fact uses the $${\mathcal S}$$-unit theorem for number fields.
##### MSC:
 11J87 Schmidt Subspace Theorem and applications 37A35 Entropy and other invariants, isomorphism, classification in ergodic theory 28D20 Entropy and other invariants
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