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Vectors with orthogonal iterates along IP-sets in unitary group actions. (English) Zbl 1027.47008

In the paper under review, the following result is proven. Suppose that \(\mu\) is a probability measure on the dual \(\widehat G\) of a countable Abelian group \(G\), \(E\) is a symmetric subset of \(G\) (\(E=-E\), \(0\in E\)) and \[ \sum_{g\in G-E,\;g\neq 0}|\widehat\mu(g)|< \delta<1/3. \] Then there exists \(f\in C^+(\widehat G)\) such that \(\|f-1\|< {3\over 2}\delta\), \((f\widehat\mu)(g)= 0\) for all \(g\in E\setminus\{0\}\) and \((f\widehat\mu)(0)= 1\).
This has the following Corollary. If \(g\mapsto U_g\) is a unitary action of a discrete Abelian group \(G\) on a Hilbert space \({\mathcal H}\) and \(\{g_i\}\subset G\) is a sequence such that both \(\{g_i\}\) and \(\{2g_i\}\) are mixing for this action, then for all \(x\in{\mathcal H}\) and \(\varepsilon> 0\) there are \(x'\overset{\varepsilon}\sim x\) (\(x'\) and \(x\) are elements of a metric space within \(\varepsilon\) of each other) and an IP-set \(\Gamma\) generated by a subsequence of \(\{g_i\}\) such that the vectors \(\{U_g x': g\in\Gamma\}\) are pairwise orthogonal. This corollary leads to simple proofs and improvements of many results in [V. Bergelson, I. Kornfeld and B. Mityagin, J. Funct. Anal. 126, 274-304 (1994; Zbl 0841.22004)] and in some other papers.
It is shown, moreover, that even in the case of a \(\mathbb{Z}\)-action, the hypothesis that \(\{g_i\}\) is mixing alone is not sufficient in the preceding corollary. Some conditions on \(G\) and on the spectrum of the action under which that assumption is sufficient are given.

MSC:

47A35 Ergodic theory of linear operators
22D10 Unitary representations of locally compact groups
47D03 Groups and semigroups of linear operators
28D05 Measure-preserving transformations
37A15 General groups of measure-preserving transformations and dynamical systems

Citations:

Zbl 0841.22004
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