  We investigate maximal abelian subalgebras (masas) in separably acting type
$II_1$ factors. We use the notion of distance between masas which we introduced
in an earlier paper in this archive, OA/0107075. The main result of the paper
is to show that for any unitary u and a masa A, the distance between A and uAu*
is comparable to the distance in 2-norm from u to the unitary normalizer N(A)
of A. In qualitative terms, this says that a unitary almost normalizes A if and
only if it is close to a normalizing unitary. Inequalities in the paper make
this statement quantitatively precise. As a consequence, all singular masas in
separably acting type $II_1$ factors are $\alpha$-strongly singular, as defined
in the above referenced paper, for a universal value of $\alpha$.
