6 problems

Dynkin's conjecture. Every compact set $S\subset {\bf R}^{n}$ carries a non-trivial doubling measure $\mu$.

Abundance conjecture. In analogy with the authors' results in the continuous setting, there are plenty of independent doubling minimizers in $\mathbb{Z}^d$ for every $d\geq 3$.

Let $1<\beta<2$ be a Pisot number other than the golden ratio, and let $\mu_{\mathbf{p}}$ be the self-similar measure considered in the paper. Non-doubling conjecture. The measure…

Assouad's doubling measure conjecture. Every complete doubling metric space has a doubling measure.

Prescribing-dimensions conjecture. Theorem also holds without the requirement that $d_*=q\Leftrightarrow D_*=q$ and $d^*=q\Leftrightarrow D^*=q$. The paper's construction does not…

Let $m$, $q$, and $s_P$ be the parameters in Theorem. Endpoint extension conjecture. Theorem also holds when $s_P=m-1$, $s_P=m$, or $s_P=q$. This would extend the theorem to the en…