Talk:Density

Asymptotic density

The article says:

Let $R=\{a/b:a,b\in A\}$ be the set of fractions of $A$. If $A$ has positive asymptotic density, then $R$ is (topologically) dense in the positive reals. If $R$ is not dense in the reals then $A$ has lower asymptotic density less than 1/2 and upper asymptotic density less than 1; and any such lower or upper density is achievable.

If $R$ is (topologically) dense in the positive reals, does it imply that $A$ has positive asymptotic density? If $P$ is the set of primes (which have null upper asymptotic density, and for which the Green–Tao theorem proved that it contains arbitrarily long arithmetic progressions), is $R=\{a/b:a,b\in P\}$ (topologically) dense in the positive reals? — Daniel Forgues 05:03, 26 October 2012 (UTC)