65 problems

Let $K_{1/3}$ be the middle-third Cantor set, let $\tau>1$, and let $W_1(\psi_\tau,0)$ denote the set of points satisfying the corresponding one-dimensional Diophantine approximati…

Let an iterated function system (IFS) generate a self-similar set satisfying the relevant hypotheses, such as the open set condition. Call the IFS non-lattice when its contraction…

Levesley–Salp–Velani conjecture.

Quasi-Assouad dimension conjecture. The quasi-Assouad dimension of $F$, and therefore its Assouad spectrum, always coincides with the Hausdorff dimension of $F$.

Let $K$ be the attractor of a self-similar iterated function system (IFS) $\{\varphi_i:i\in\Lambda\}$ on $\mathbb{R}$ that contains no exact overlaps. Let $\lambda_i$ be the contra…

Let $E,F\subset\mathbb{R}^d$ be two totally disconnected non-trivial self-similar sets, generated by iterated function systems $\Phi=\{\varphi_i\}_{i\in I}$ and…

Exact overlap conjecture. If there is a dimension drop for $\Phi$, then $\Phi$ has exact overlaps.

Let a self-similar set be generated by a finite iterated function system of similarities, and let its similarity dimension be the value defined by the usual similarity-dimension eq…

Let $F$ and $E$ be self-similar sets, and suppose that $F$ embeds into $E$ by an affine map. Feng–Huang–Rao's algebraic-dependence conjecture. Every contraction ratio of $F$ should…

Pisot characterization conjecture. One has $l_n(q)>0$ if and only if $q$ is a Pisot number.

Benjamini–Solomyak pair-correlation conjecture. For almost every $\lambda\in(1/2,1)$, there exist $c,C>0$ such that

Let $\phi_i(x):=x/5+(i-1)/5$ for $i\in\{1,\ldots,5\}$. Let $M$ and $M'$ be the attractors of the IFSs $\{\phi_1,\phi_3,\phi_5\}$ and $\{\phi_1,\phi_4,\phi_5\}$, respectively. The s…

Let $K\subset\mathbb{R}^{d}$ be a compact set, and let $\mathcal{M}(K)$ be the space of probability measures on $K$ with its weak topology. For a measure $\mu\in\mathcal{M}(K)$, wr…

Let a p.c.f. self-similar set mean a post-critically finite self-similar set, and let a finitely ramified self-similar set be a self-similar set with finitely ramified cell structu…

For $\lambda$ in the open unit disk, let … be the associated self-similar set, and let $\nu_\lambda$ be the corresponding complex Bernoulli convolution. In the case…

Let $n=4$ and let $r$ satisfy $0<r<R_4=1/2$. Then the regular $4$-flake dust is the Cantor dust $\mathcal{F}^{4,r}=C^{1/r}$. The intermediate-ratio Cantor-dust conjecture. For…

Super-exponential condensation conjecture. Any analytic IFS which has super-exponential condensation but no exact overlaps must be sub-conjugated to a self-similar IFS.

Let $C^r$ denote the self-similar fractal family defined in the paper, and let Minkowski measurability mean existence of a finite, positive Minkowski content. The conjecture for…

Let an iterated function system generate a self-similar fractal set satisfying the open set condition. An IFS is non-lattice when its similarity scaling ratios satisfy the correspo…

Let $K$ be a self-similar set in the plane, and call a direction $\pi\in G(2,1)$ exceptional for orthogonal projections of $K$ if the projection dimension equality does not hold. C…

Arithmetic-product conjecture. The conclusion above should hold without any rational-incommensurability assumption on the self-similar sets.

Feng–Huang–Rao's conjecture. If, for some $i\in I$, $\log\alpha_i\notin\operatorname{span}_{\mathbb{Q}}\\{\log\beta_j\\}_{j\in J}$, then

Logarithmic commensurability conjecture. The contraction ratios are logarithmically commensurable, meaning

Let $\Phi$ be a self-similar iterated function system on $\mathbb R^2$ and let $K$ be its attractor. The Luo–Rao–Xiong local connectedness conjecture. If $\Phi$ satisfies the open…

The symbolic subsequence conjecture. There always exists an increasing sequence of natural numbers $\{c_n(\mathbf{t})\}_n$, dictated by the symbolic structure of $\mathbf{t}$, such…