Additive relations in fields: an entropy approach (Q1896591)

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 \textit{A. J. van der Poorten} and \textit{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.