  In this paper we provide asymptotic upper bounds on the complexity in two
(closely related) situations. We confirm for the total doubling coverings and
not only for the chains the expected bounds of the form $$ \kappa({\mathcal U})
\le K_1(\log ({1}/{\delta}))^{K_2} . $$ This is done in a rather general
setting, i.e. for the $\delta$-complement of a polynomial zero-level
hypersurface $Y_0$ and for the regular level hypersurfaces $Y_c$ themselves
with no assumptions on the singularities of $P$. The coefficient $K_2$ is the
ambient dimension $n$ in the first case and $n-1$ in the second case. However,
the question of a uniform behavior of the coefficient $K_1$ remains open.
  As a second theme, we confirm in arbitrary dimension the upper bound for the
number of a-charts covering a real semi-algebraic set $X$ of dimension $m$ away
from the $\delta$-neighborhood of a lower dimensional set $S$, with bound of
the form $$ \kappa(\delta) \le C (\log ({1}/{\delta}))^{m} $$ holding uniformly
in the complexity of $X$. We also show an analogue for level sets with
parameter away from the $\delta$-neighborhood of a low dimensional set. More
generally, the bounds are obtained also for real subanalytic and real
power-subanalytic sets.
