6 problems

Let $A_2,A_3\subset[0,1]$ be closed sets invariant under multiplication by $2$ and $3$ modulo $1$, respectively, and suppose … For $g(x)=\exp(-(-\log x)^s)$, Hausdorff-measure conj…

Furstenberg intersection conjecture. Every point of $A_2\cap A_3$ is rational. This is posed as a strengthening of known dimension and sparsity results for intersections of…

Let $T_a$ be the fixed transformation, and let $f$ be a real-analytic function on either the real line or the circle: … Assume that $f$ is not affine. A set is jointly $T_a$- and…

Symmetric invariant-set conjecture. At every $\varepsilon_n$, new symmetric invariant sets appear; these sets are contained in $\mathcal{S}$ and are symmetric. The conjecture is mo…

Second asymmetric-set conjecture. There exists such a polyhedron $P_3$ for which $\mathcal{A}_2$ is symmetric with respect to $S_1,S_2$ and $S_4$, asymmetric with respect to…

Let $f:\overline{{\mathbb R}^n}\to\overline{{\mathbb R}^n}$ be quasiregular with $\operatorname{deg}(f)>K_I(f)$. Let $E(f)$ be the exceptional set. A set $X$ is completely invarian…