15 problems

Let $C$) be the middle-third Cantor set, let $\mu$ be its natural Cantor probability measure, and write … For a positive function $\psi$, define … Equivalently, $x\in W_2(\psi)$ if…

Let $C$ be the middle-third Cantor set and let … W2:=left{xin[0,1]: |2^nx|<(n)text{ for infinitely many }ninright}. … is monotonic, then … This is a Khintchine-type zeroone law for…

Cover-time conjecture. Almost surely,

Let $\sigma$colon be a minimal homeomorphism of the Cantor set. An action is biminimal if, for every pair of points with open neighbourhoods $x\in\mathcal U$ and $y\in\mathcal V$,…

Let $K_0$ be the Cantor ternary set and let $\nu_{K_0}$ be its Hausdorff measure for $h(t)=t^{\frac{\log 2}{\log 3}}$. Let $W_n(\nu_{K_0})$ denote the associated sequence of quanti…

Let $E$ be a closed subset of the Cantor set, and let $\operatorname{Homeo}(E)$ denote its homeomorphism group. A basic counterexample to recovering the homeomorphism type of $E$ f…

Let $K$ be the middle-third Cantor set, let $\kappa=\operatorname{dim_{H}}K$, and let $w_n^{\ast}(\xi)$ be the exponent governing approximation of $\xi$ by algebraic numbers of deg…

Non-triadic-base rational-counting conjecture. For all $\varepsilon>0$,

Approximation-counting conjecture near the Cantor set. Combined with the density conclusion for rational points lying in $K$, the preceding heuristic arguments imply

Let $K$ be the middle-third Cantor set, let $\mathcal{P}$ be the set of reduced rational pairs $(p,q)$ used in the paper, and let…

Let $K$ be the middle-third Cantor set, let $\kappa=\operatorname{dim_{H}}K$, let $b$ be an integer that is not a power of three, and let $v_b(\xi)$ be the approximation exponent a…

Long-period conjecture. For every $K>2/\log(3/2)$, one has $S_n^{(K)}=S_n$ for all sufficiently large $n$; in particular,

Beresnevich–Frolenkov–Rynne conjecture. For every $\varepsilon_1>0$,

Subquadratic rational-counting conjecture. There exists a constant $\gamma<2$ such that

Let $C$ be the standard Cantor set, and let … Here $\gcd(p,q)=1$ means that the rational is written in reduced form. Mahler's rational-counting conjecture. For every $\epsilon>0$,…