The classical Diophantine approximation problem studies solutions $(p,q) \in \mathbb{Z} \times \mathbb{N}$ to inequalities roughly of the form $$\qquad \qquad \qquad \qquad \qquad \qquad \left|\alpha - \frac{p}{q}\right|<\frac{1}{q^\lambda} \qquad \qquad \qquad \qquad \qquad \qquad \qquad(1)$$
for some real $\lambda \geq 1$, where $\alpha$ is real. It's well-known that "most" irrational $\alpha$ can be approximated reasonably well in this manner (for most $\alpha$, we have infinitely many solutions to $(1)$ when $\lambda=2$) but not too well (for most $\alpha$, we only have finitely many solutions if $\lambda>2$).
One thing we can notice here is that in the classical problem, we take $p$ to be integral. In this question, I'm interested in altering the possible choices for $p$. Instead of necessarily taking $p \in \mathbb{Z}$, we take $p \in S$, for some fixed infinite $S \subset \mathbb{R}$. For instance, we can ask, what if we allow in addition $p=\sqrt{2}$? In other words, we are interested in approximating reals by the rationals, as well as numbers of the form $\frac{\sqrt{2}}{q}$?
$\sqrt{2}$ by itself will not make a difference. Allowing one additional value for $p$ is immaterial, and it will not affect the theory significantly (except possibly for the fact $\sqrt{2}$ is now trivially perfectly-approximable). However, allowing infinitely many more values may. For instance, say we let $p \in \mathbb{Q}$ (whereas originally, $p \in \mathbb{Z}$). Then, it is not hard to see that $(1)$ will have infinitely many solutions for any choice of $\alpha, \lambda$.
I conjecture that the crucial thing in this regard is the density of the set of $p$ we choose from.
$\textbf{Problem 1}$: Let $S \subset \mathbb{R}$. I conjecture that the following are equivalent.
$(1)$: $S$ is dense as a subset of $\mathbb{R}$.
$(2)$: For any $\lambda>0$ and any $\alpha \in \mathbb{R}$, there are infinitely many solutions to $\left|\alpha - \frac{p}{q}\right|<\frac{1}{q^\lambda}$ where $(p,q) \in S \times \mathbb{N}$.
Here, $S$ takes the role of the "$p$-set".
Now, if $S$ is a discrete subset of $\mathbb{R}$ unbounded above and below, similar to $\mathbb{Z}$, then things are less clear, and I genuinely do not have a clue how the theory might change.
$\textbf{Problem 2}$: Is there a detailed study of how the theory changes if we change the set $S$ from which we take take numerators? How precisely do quantities, like, say, the irrationality measure, change? Is there an all-encompassing fundamental theorem, in the general case, similar to, say, the Duffin–Schaeffer conjecture?
I have a general feeling that if $S$ is a very sparse subset of integers, say something like $S=\{\pm 2^n : n \in \mathbb{N}\}$, then it's likely that things go awry, and even a theorem like Dirichlet's approximation theorem will fail. The reason I tend to feel this way is that Duffin–Schaeffer conjecture (which, I should note, is now a theorem, see this), implies that if the denominators are chosen quite "sparsely", then things go awry, so I would not be surprised if a similar claim holds for the numerators.
This is possibly a very broad question, so references are welcomed.