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Equidistribution in the dual group of the \(S\)-adic integers. (English) Zbl 1349.11104

Summary: Let \(X\) be the quotient group of the \(S\)-adele ring of an algebraic number field by the discrete group of \(S\)-integers. Given a probability measure \(\mu \) on \(X^d\) and an endomorphism \(T\) of \(X^d\), we consider the relation between uniform distribution of the sequence \(T^n\mathbf{x}\) for \(\mu\)-almost all \(\mathbf{x}\in X^d\) and the behavior of \(\mu\) relative to the translations by some rational subgroups of \(X^d\). The main result of this note is an extension of the corresponding result for the \(d\)-dimensional torus \(\mathbb T^d\) due to B. Host.

MSC:

11J71 Distribution modulo one
11K06 General theory of distribution modulo \(1\)
54H20 Topological dynamics (MSC2010)
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References:

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