Density inequalities for sets of multiples (Q1907840)

For a finite set \(A\) of integers let \(M(A)\) be the set of natural numbers divisible by at least one element of \(A\), and let \({\mathbf d}\) denote asymptotic density. For two sets \(A\), \(B\) write \[ A\times B= \{ab\}, \qquad [A, B]= \{[a, b]\}, \qquad (A, B)= \{(a, b)\}, \] where \(a\in A\), \(b\in B\). A classical theorem of \textit{F. A. Behrend} [Bull. Am. Math. Soc. 54, 681-684 (1948; Zbl 0031.34602)] can be reformulated as \({\mathbf d} M[A, B]\geq {\mathbf d} M(A) {\mathbf d} M(B)\). Here this inequality is sharpened to \({\mathbf d} M(A, B){\mathbf d} M[A, B]\geq {\mathbf d} M(A) {\mathbf d} M(B)\), and the new inequality \[ {\mathbf d} M(A\times B)\leq {\mathbf d} M(A) {\mathbf d} M(B) \] is established.