For every compact set E in the real line with dim_A E > 0, there exist measures supported on E with any prescribed upper Assouad dimension D > dim_A E, and for any d < dim_L E, a measure with lower Assouad dimension d.
On the upper regularity dimensions of measures
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abstract
We study the \emph{upper regularity dimension} which describes the extremal local scaling behaviour of a measure and effectively quantifies the notion of \emph{doubling}. We conduct a thorough study of the upper regularity dimension, including its relationship with other concepts such as the Assouad dimension, the upper local dimension, the $L^q$-spectrum and weak tangent measures. We also compute the upper regularity dimension explicitly in a number of important contexts including self-similar measures, self-affine measures, and measures on sequences.
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math.CA 1years
2019 1verdicts
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Measures with specified support and arbitrary Assouad dimensions
For every compact set E in the real line with dim_A E > 0, there exist measures supported on E with any prescribed upper Assouad dimension D > dim_A E, and for any d < dim_L E, a measure with lower Assouad dimension d.