Additivity of the ideal of microscopic sets

A set M⊂RM⊂R is microscopic if for each ε>0ε>0 there is a sequence of intervals (Jn)n∈ω(Jn)n∈ω covering M   and such that |Jn|≤εn+1|Jn|≤εn+1 for each n∈ωn∈ω. We show that there is a microscopic set which cannot be covered by a sequence (Jn)n∈ω(Jn)n∈ω with {n∈ω:Jn≠∅}{n∈ω:Jn≠∅} of lower asymptotic density zero. We prove (in ZFC) that additivity of the ideal of microscopic sets is ω1ω1. This solves a problem of G. Horbaczewska. Finally, we discuss additivity of some generalizations of this ideal.