The spectral boundary of complemented invariant subspaces in \(L^p(\mathbb{R})\) (Q2764653)
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scientific article; zbMATH DE number 1690773
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | The spectral boundary of complemented invariant subspaces in \(L^p(\mathbb{R})\) |
scientific article; zbMATH DE number 1690773 |
Statements
21 September 2003
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spectral subboundary
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arithmetical thickness
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Hausdorff dimension
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arithmetical progressions
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The spectral boundary of complemented invariant subspaces in \(L^p(\mathbb{R})\) (English)
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For a compact set in \(\mathbb{R}\), the authors introduce the notion of ``arithmetical thickness'' \((A\)-thickness) and construct some kind of compact set. Their main result is as follows:NEWLINENEWLINENEWLINETheorem. There exists a compact perfect set \(K\subset\mathbb{R}\) such that (i) \(\dim_H K=0\) and (ii) \(K\) is \(A\)-thick, where \(\dim_HK\) denotes the Hausdorff dimension of \(K\). -- They also show that a compact set in \(\mathbb{R}\) which is \(A\)-thick contains arbitrarily long arithmetical progressions.
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