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Gerhard, I'd eagerly await the follow ups you've mentioned regarding this answer, as well as the remarks you mention on the arXiv preprint which inspired your post. Thank you!
Hi Daniel, interesting question. It seems that at least one nice thing that can be said is that the primes in such a multiplicatively closed set must have zero density in the primes. This would be due to Theorem 1 of this paper (arxiv.org/abs/1110.0708) of Moree, and also from this MO question (mathoverflow.net/questions/94543/density-of-a-set-of-integers), Density of a set of integers. Wonder exactly how sparse they have to be, though.
Hi @js, and thanks for your comment. It indeed seems elusive. And from the comments in the first question linked to here, it seems that even the question of which integers are represented by multivariate polynomials is not all that straightforward...
