Additivity of the ideal of microscopic sets

摘要

A set M subset of R is microscopic if for each epsilon > 0 there is a sequence of intervals (J(n))(n is an element of w) covering M and such that vertical bar J(n)vertical bar <= epsilon(n+1) for each n is an element of w. We show that there is a microscopic set which cannot be covered by a sequence (J(n))(n is an element of w) with {n is an element of w : J(n) not equal theta}of lower asymptotic density zero. We prove (in ZFC) that additivity of the ideal of microscopic sets is w(1). This solves a problem of G. Horbaczewska. Finally, we discuss additivity of some generalizations of this ideal. (C) 2016 Elsevier B.V. All rights reserved.