Compact groups all elements of which are almost right Engel

We say that an element $g$ of a group $G$ is almost right Engel if there is a finite set $\mathcal{R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[\mathrm{.}\mathrm{.}\mathrm{.}[[g,x],x],\dots ,x]$ belong to $\mathcal{R}(g)$, that is, for every $x\in G$ there is a positive integer $n(x,g)$ such that $[\mathrm{.}\mathrm{.}\mathrm{.}[[g,x],x],\dots ,x]\in \mathcal{R}(g)$ if $x$ is repeated at least $n(x,g)$ times. Thus, $g$ is a right Engel element precisely when we can choose $\mathcal{R}(g)=\{1\}$. We prove that if all elements of a compact (Hausdorff) group $G$ are almost right Engel, then $G$ has a finite normal subgroup $N$ such that $G\mathrm{/}N$ is locally nilpotent. If in addition there is a uniform bound $|{\mathscr R}(g)|\leq m$fortheordersofthecorrespondingsets,thenthesubgroup$N$canbechosenoforderboundedintermsof$m$. The proofs use the Wilson–Zelmanov theorem saying that Engel profinite groups are locally nilpotent and previous results of the authors about compact groups all elements of which are almost left Engel.