On profinite groups with many elements with large nilpotentizer and generalizations
Let $F$ be a family of finite groups closed under taking subgroups, quotients and finite direct products. Given an element $g$ of a profinite group $G$, the $F$-izer of $g$ in $G$ is the set of elements $x$ in $G$ such that $\langle g,x\rangle$ is a pro-$F$-group. Let $F(G)$ be the set of elements $g$ of $G$ such that the $F$-izer of $g$ in $G$ has positive Haar measure.