On the self-similarity problem for Gaussian-Kronecker flows (Q2855906)

The topic of the article is the realization of countable subgroups as groups of self-similarities of a measure preserving flow. Let \({(X, \mathcal{B},\mu)}\) be a standard probability Borel space and \({\mathcal{T}}\) be Gaussian-Kronecker flow. This means that:NEWLINENEWLINE1) there is a \({\mathcal{T}}\)-invariant subspace \({\mathcal{H}\subset L^2_0(X,\mathcal{B},\mu)}\) consisting of Gaussian variables and \({\sigma(\mathcal{H})=\mathcal{B}},\) andNEWLINENEWLINE2) the maximal spectral type measure of \(\mathcal{T}\) is a continuous Kronecker measure (its support is the union of an increasing sequence of Kronecker sets [\textit{M. Lemańczyk} and \textit{F. Parreau}, Proc. Am. Math. Soc. 127, No. 7, 2073--2081 (1999; Zbl 0923.28007)]). NEWLINENEWLINENEWLINEGiven a countable multiplicative subgroup \({H\subset\mathbb{R}_{+}^*},\) define \({G=-H\cup H}.\) The main result of the article states that the group \(G\) can be realized as the set of self-similarities of \(\mathcal{T}\) if and only if \(H\) is an additively \(\mathbb{Q}\)-independent set.