Hausdorff measures of subgroups of \mathbb{R }/\mathbb{Z } and \mathbb{R } |
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Authors: | Hans Weber Enrico Zoli |
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Institution: | 1. Dipartimento di Matematica ed Informatica, Università degli Studi di Udine, via delle Scienze 206, Udine, 33100, Italy 2. I.T.I.P. “Bucci”, via Nuova 45, Faenza, 48018, Italy
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Abstract: | For a sequence $\underline{u}=(u_n)_{n\in \mathbb{N }}$ of integers, let $t_{\underline{u}}(\mathbb{T })$ be the group of all topologically $\underline{u}$ -torsion elements of the circle group $\mathbb{T }:=\mathbb{R }/\mathbb{Z }$ . We show that for any $s\in ]0,1$ and $m\in \{0,+\infty \}$ there exists $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has Hausdorff dimension $s$ and $s$ -dimensional Hausdorff measure equal to $m$ (no other values for $m$ are possible). More generally, for dimension functions $f,g$ with $f(t)\prec g(t), f(t)\prec \!\!\!\prec t$ and $g(t)\prec \!\!\!\prec t$ we find $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has at the same time infinite $f$ -measure and null $g$ -measure. |
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