9 problems

Let $k$ be a positive integer, partition the range $[0,1]$ uniformly into $k$ pieces, and choose an appropriate piecewise linear base function. Uniform-partition extension conjectu…

Let $I=[0,1]$ be the domain, let $S$ be the vertical scaling function, and let $M_k$ and $M'_k$ be the matrices associated with the $k$-th subdivision, with spectral radii denoted…

Let $F$ be a self-affine set, allowing overlaps among the defining affine pieces. Its upper and lower box dimensions are denoted by $\overline{\dim}_B F$ and $\underline{\dim}_B F$…

Box-dimension conjecture. Any trajectory of $X_{m,n}$ near the origin $(x,y)=(0,0)$ is Minkowski nondegenerate and has box dimension

Principal-projection box-dimension conjecture. Both projection box dimensions should exist:

Graph-directed self-affine box-dimension conjecture. The box dimension of $E$ exists, so that

Let $F_C\subset\mathbb R$ be an inhomogeneous self-similar set, with homogeneous attractor $F_\emptyset$ and condensation set $C$. Write $\overline{\dim}_{\mathrm B}$ for upper box…

Box-dimension equality conjecture. The equality holds in the displayed lower-bound formula: $\dim_B(L_a)$ equals the corresponding value in the cases listed there.

Upper graph box dimension dichotomy conjecture. The upper graph box dimension of $X$ is either equal to the upper box dimension of $X$ plus one or equal to one: