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Question 1: The set is dense.

Suppose that we are given a fixed $x\in\mathbb{R}$. Then let $p$ be a large prime. If $p$ is sufficiently large, then there will be a prime $$q\in\left[px,\ px+\left(px\right)^{0.525}\right]$$ by the work of Baker, Harman and Pintz on prime gaps. This implies that $$\left|x-\frac{q}{p}\right|\ll_x p^{-0.475},$$ which becomes arbitrarily small as we take $p\rightarrow\infty$. This proves that for any $\epsilon>0$, there exists $p,q$ such that $\left|x-\frac{q}{p}\right|\leq \epsilon.$

Question 2: We can find infinitely many solutions to $$1\leq qp-rs\leq a$$ for primes $p,q,r,s$ and all $a\geq 26$. Under the Elliott-Halberstam Conjecture, we can take $a\geq 6$.

This is a corollary of the work of Goldston, Graham, Pintz and Yıldırım on the gaps between almost primes. They prove that if $q_n$ is the $n^{th}$ almost prime, then $$\liminf_{n\rightarrow \infty} q_{n+1}-q_n \leq 26,$$ and that the upper bound may be reduced to $6$ under the Elliott-Halberstam Conjecture. Since $q_n=pq$ and $q_{n+1}=rs$ where $p,q,r,s$ are primes, this yields the above claim.

Edit: The more recent work of Goldston, Graham, Pintz and Yıldırım show that we can take $a=6$ unconditionally. (Thank you to quid for mentioning this in the comments)

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Question 1: The set is dense.

Suppose that we are given a fixed $x\in\mathbb{R}$. Then let $p$ be a large prime. If $p$ is sufficiently large, then there will be a prime $$q\in\left[px,\ px+\left(px\right)^{0.525}\right]$$ by the work of Baker, Harman and Pintz on prime gaps. This implies that $$\left|x-\frac{q}{p}\right|\ll_x p^{0.475},$$ p^{-0.475},$$which becomes arbitrarily small as we take p\rightarrow\infty . This proves that for any \epsilon>0, there exists p,q such that \left|x-\frac{q}{p}\right|\leq \epsilon. Question 2: We can find infinitely many solutions to$$1\leq qp-rs\leq a$$for primes p,q,r,s and all a\geq 26. Under the Elliott-Halberstam Conjecture, we can take a\geq 6. This is a corollary of the work of Goldston, Graham, Pintz and Yıldırım on the gaps between almost primes. They prove that if q_n is the n^{th} almost prime, then$$\liminf_{n\rightarrow \infty} q_{n+1}-q_n \leq 26,$$and that the upper bound may be reduced to 6 under the Elliott-Halberstam Conjecture. Since q_n=pq and q_{n+1}=rs where p,q,r,s are primes, this yields the above claim. 4 deleted 59 characters in body; edited body Question 1: The set is dense. Suppose that we are given a fixed x\in\mathbb{R}. Then let p be a large prime. If p is sufficiently large, then there will be a prime$$q\in\left[px,\ px+\left(px\right)^{0.525}\right]$$by the work of Daniel A. GoldstonBaker, Janos Pintz Harman and Cem Y. Yıldırım Pintz on prime gaps. This implies that$$\left|x-\frac{q}{p}\right|\ll_x p^{0.475},$$which becomes arbitrarily small as we take p\rightarrow\infty . This proves that for any \epsilon>0, there exists p,q such that \left|x-\frac{q}{p}\right|\leq \epsilon. Question 2: We can find infinitely many solutions to$$1\leq qp-rs\leq a$$for primes p,q,r,s and all a\geq 26. Under the Elliott-Halberstam Conjecture, we can take a\geq 6. This is a corollary of the work of Daniel A. Goldston, Sidney W. Graham, Janos Pintz and Cem Y. Yıldırım on the gaps between semi almost primes. They prove that if q_n is the n^{th} almost prime, then$$\liminf_{n\rightarrow \infty} q_{n+1}-q_n \leq 26, and that the upper bound may be reduced to $6$ under the Elliott-Halberstam Conjecture. Since $q_n=pq$ and $q_{n+1}=rs$ where $p,q,r,s$ are primes, this yields the above claim.