8 problems

Throughout, let $K$ be the middle-third Cantor set, let $\mu$ be its natural probability measure, and for $\psi:\mathbb R\to[0,\infty)$ and $y\in\mathbb R$ define … Here $\|x\|$ de…

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…

Let $\psi:{\mathbb{N}}\to{\mathbb{R}}^+$ be monotonic, and let … where $\mathcal A(2)=\{2^n:n=0,1,2,\ldots\}$ and $\mu$ denotes Lebesgue measure. Velani's conjecture. … This is the…

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…

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,

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 $\epsilon>0$,…