  A set $M\subset\mathbb{R}$ is microscopic if for each $\varepsilon>0$ there
is a sequence of intervals $(J_n)_{n\in\omega}$ covering $M$ and such that
$|J_n|\leq \varepsilon^{n+1}$ for each $n\in\omega$. We show that there is a
microscopic set which cannot be covered by a sequence $(J_n)_{n\in\omega}$ with
$\{n\in\omega:J_n\neq\emptyset\}$ of lower asymptotic density zero. We prove
(in ZFC) that additivity of the ideal of microscopic sets is $\omega_1$. This
solves a problem of G. Horbaczewska. Finally, we discuss additivity of some
generalizations of this ideal.
