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Attainable forms of intermediate dimensions. (English) Zbl 1509.28005

Summary: The intermediate dimensions are a family of dimensions which interpolate between the Hausdorff and box dimensions of sets. We prove a necessary and sufficient condition for a given function \(h(\theta)\) to be realized as the intermediate dimensions of a bounded subset of \(\mathbb{R}^d\). This condition is a straightforward constraint on the Dini derivatives of \(h(\theta)\), which we prove is sharp using a homogeneous Moran set construction.

MSC:

28A78 Hausdorff and packing measures
28A80 Fractals
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