Measures with specified support and arbitrary Assouad dimensions

Hare, Kathryn; Mendivil, Franklin; Zuberman, Leandro

doi:10.1090/proc/15001

Measures with specified support and arbitrary Assouad dimensions
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by Kathryn E. Hare, Franklin Mendivil and Leandro Zuberman PDF

Proc. Amer. Math. Soc. 148 (2020), 3881-3895 Request permission

Abstract:

We show that if the upper Assouad dimension of the compact set $E\subseteq \mathbb {R}$ is positive, then given any $D>\dim _{A}E$ there is a measure with support $E$ and upper Assouad (or regularity) dimension $D$. Similarly, given any $0\leq d<\dim _{L}E,$ there is a measure on $E$ with lower Assouad dimension $d$.

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