  Given an arbitrary d>0 we construct a group G and a group ring element S in
Z[G] such that the spectral measure mu of S has the property that mu((0,eps)) >
C/|log(eps)|^(1+d) for small eps. In particular the Novikov-Shubin invariant of
any such S is 0. The constructed examples show that the best known upper bounds
on mu((0,eps)) are not far from being optimal.
