Can integers be distorted to make primes more regular?

Given a set $P$ of real numbers $\ge 1$, define the gap among different products in $P$ as

$$g(P) = \inf \big\{\prod_{i=1}^n p_i^{a_i} - \prod_{i=1}^n p_i^{b_i} \mid p_i\in P;\,\, p_i\ne p_j \,\text{ if }\, i\ne j ; \,\,a_i, b_i \in \mathbb{N}\cup \{0\};\,\, a_1 \ne b_1 \big\}$$

If $g(P)>0$ in one could say that the set of products of members of $P$ has unique factorization in an essentially strong way.

The main question is: can any infinite sets $P$ with $g(P)> 0$ be produced with a density similar to that of the primes, but more regularly distributed (for example $p_i=(i+a)\log(bi+c)+d$ for some constants $a,b,c,d$, or $iH_i+a$)?

Can any interesting set $P$ with $g(P)>0$ be produced at all?

• – Terry Tao Mar 31 '15 at 16:47
• if $P$ is finite with $n$ elements, it's pretty easy to see that $g(P)$ is 0 in a dense subset of $[1,\infty)^n$. I think it should be possible to show that $g(P)>0$ in a dense subset as well, using the sparsity of products generated by $P$ in $\mathbb{R}$. – Yaakov Baruch Mar 31 '15 at 17:00