Additivity of the ideal of microscopic sets

A set $M\subset \mathbb{R}$ is microscopic if for each $\epsilon >0$ there is a sequence of intervals $(J_n{)}_{n\in \omega }$ covering $M$ and such that $\mathrm{\mid }{J}_n\mathrm{\mid }\le {\epsilon }^{n+1}$ 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 $\{n\in \omega :J_n\mathrm{\ne }\mathrm{\varnothing }\}$ 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.