It is well known that for every real number we can find a rational number arbitrarily close to it. In other words:
Let $a\in\mathbb{R}$. For each $\epsilon\gt 0$, there exist $p,q\in\mathbb{Z}$ such that $\left|a-\frac{p}{q}\right|\lt\epsilon$.
But can we let $p$ belong to some infinite subset of $\mathbb{Z}$ so that this is not true? In other words:
Does there exist an infinite subset $S\subset\mathbb{Z}$ such that there exist $a\in\mathbb{R}$ and $c>0$ so that for every $p\in S$ and $q\in\mathbb{Z}$, we have $c\lt\left|a-\frac{p}{q}\right|$?
What if we say $q$ is also in $S$?
If such subsets exist I’d naturally want to know the largest one.
With $q \in S$ we can say even more. We can find an infinite $S$ such that for any given $c \gt 0$ we can find $a$ such that $\forall p,q \in S, |a-\frac pq| \gt c$ Just take $S$ to be the powers of $2$. Then $\frac pq$ is also a power of $2$, so given $c$ we just take $a$ large enough that powers of $2$ are far enough apart.
If we don't insist that $q \in S$ this fails. As $S$ is infinite there are arbitrarily large elements. The distance between $\frac pq$ and $\frac p{q+1}$ is about $\frac p{q^2}$, which is about $\frac aq$ and can be made arbitrarily small.