Multiplicative representations of integers

Lehh ≧ 2, and let ?=(B ₁, …,B _h ), whereB ₁ ? N={1, 2, 3, …} fori=1, …,h. Denote by g_?(n) the number of representations ofn in the formn=b ₁ …b _h , whereb _i ∈B _i . If v (n) > 0 for alln >n ₀, then ? is anasymptotic multiplicative system of order h. The setB is anasymptotic multiplicative basis of order h ifn=b ₁ …b _n is solvable withb _i ∈B for alln >n ₀. Denote byg(n) the number of such representations ofn. LetM(h) be the set of all pairs (s, t), wheres=lim g_? (n) andt=lim g_? (n) for some multiplicative system ? of orderh. It is proved that {fx129-1} In particular, it follows thats ≧ 2 impliest=∞. A corollary is a theorem of Erdös that ifB is a multiplicative basis of orderh ≧ 2, then lim g_? g(n)=∞. Similar results are obtained for asymptotic union bases of finite subsets of N and for asymptotic least common multiple bases of integers.