If $a$ is any real irrational, then the set of numbers of the form $ax+y$ with $x$ and $y$ co-prime integers is dense in $\mathbb{R}$. I managed to prove this, in what I suspect is an overly complicated way, in response to this question.
I think there must be a more perspicuous proof of this fundamental fact. Furthermore, I expect that the following more general assertion is true:
If $f\in \mathbb{R}[x]$ is any polynomial at least one of whose coefficients other than the constant term is irrational, then the numbers of the form $f(x)+y$ with $x$ and $y$ co-prime integers are dense in $\mathbb{R}$.
Now it seems to me that this ought to be a very well known fact or open problem (or maybe it is false!) but I can't find a single reference to it. Can anyone provide a proof or a reference?