Almost Engel compact groups

We prove that if a compact (Hausdorff) group G is almost Engel, then G has a finite normal subgroup N such that G/N is locally nilpotent. If in addition there is a uniform bound |E(g)|⩽m for the orders of the corresponding sets, then the subgroup N can be chosen of order bounded in terms of m. The proofs use the Wilson-Zelmanov theorem saying that Engel profinite groups are locally nilpotent.