12 problems

Density conjecture. The real parts of the complex dimensions of $\mathcal{L}$ form a set that is dense in the connected interval $[\sigma_l,D]$.

Work in the Heisenberg group $\mathbb{H}$, equipped with its standard metric and Hausdorff measure, and consider line segments lying entirely along the $x$- or $y$-axis. The dimens…

Let $(A,\Omega)$ be a relative fractal drum, and suppose that the non-real complex dimensions of $(\infty,\Omega)$ arise from the unbounded set $\Omega$. Say that $\Omega$ is metri…

Self-similarity fractality conjecture. The definition of fractality based on the possession of a nonreal complex dimension, understood as a form of self-similarity, is a step forwa…

Maximal-complex-dimension conjecture. The first term at which the asymptotic expansion of $V_A(\varepsilon)$ fails to be monotonic is directly connected to the nonreal complex dime…

Self-similar-set fractality conjecture. Analogous results to those established for self-similar sprays should hold for the scaling complex dimensions of $A$; in particular, every s…

Nongeneric interval conjecture. In the nongeneric case, the set of such dimensions $d$ should be dense in finitely many compact nonempty intervals, with more than one interval.

Principal-complex-dimension conjecture. The presence of a nonreal principal complex dimension is equivalent to $A$ not being Minkowski measurable. Equivalently, $A$ is Minkowski me…

Cantor graph complex-dimension conjecture. One should have

Complex dimensions of Cartesian products. One should have

Let $A$ be the compact set considered above, let $\zeta_A$ be its distance zeta function, and for each $k\in\mathbb{Z}$ set … where $\mathbf{p}=2\pi/\log 3$. The residue…

Let $K$ be the fractal under consideration, and assume the conditions of Theorems and. A complex dimension is called nontrivial when it is one of the complex dimensions considered…