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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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