Random Duffin–Schaeffer conjecture

Let $f:\mathbb{N}\to\mathbb{N}_0$ satisfy $0\leqslant f(n)\leqslant n$, let $\Omega_f$ be the associated random space of numerator sets, and let $W^P(\Psi)$ denote the set of numbers approximable by the fractions whose numerators belong to the random choice $P\in\Omega_f$. In the case $f=\varphi$, write $\mathbb{P}_\varphi$ for the corresponding probability measure. Random Duffin–Schaeffer conjecture. For $\mathbb{P}_\varphi$-almost every $P$, the random Khintchine property holds without the monotonicity assumption: $W^P(\Psi)$ has full measure when the associated sum in property (rkt) diverges and measure zero when it converges. This is the generic random analogue of the Duffin–Schaeffer conjecture; the paper notes that it has no obvious implication relationship with the classical conjecture, although it would imply Catlin's conjecture.