2 problems

Khintchine's moving-target conjecture. If $\psi:\mathbb{N}\tomathbb{R}_{\geq 0}$ is decreasing and $\sum_{q=1}^{\infty}\psi(q)=\infty$, then

Let $T_2:\mathbb{R}/\mathbb{Z}\to\mathbb{R}/\mathbb{Z}$ be the doubling map, let $\mu$ denote Lebesgue measure on $\mathbb{R}/\mathbb{Z}$, let $\psi:\mathbb{N}\to[0,\infty)$, and l…