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