TITLE: Distribution of polynomials mod 1 using co-prime integers
QUESTION [9 upvotes]: 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?

REPLY [3 votes]: The continued fraction for $a$ gives rational approximations $$\frac{p_i}{q_i} \approx a$$ with $|q_ia-p_i| \lt \frac1{q_i}$. Furthermore,  successive approximants satisfy $p_{i+1}q_q-p_iq_{i+1}=\pm1.$ 
Fix an index $i$ and consider the sequence of expressions $$r_k=x_ka+y_k=(q_{i-1}+kq_{i})a-(p_{i-1}+kp_{i})$$ where $k \in \mathbb{Z}$. The values $r_k$ form a two way infinite arithmetic progression with common difference  $q_ia-p_i \lt \frac{1}{q_{i}}$ and thus allow us to approximate any desired real to an accuracy of better than $\frac{1}{2q_i}$. It remains only to note that 

The denominators $q_i$ increase (exponentially) and thus allow us to get any accuracy we desire.
$x_{k+1}y_k-x_{k}y_{k+1}=p_{i+1}q_i-p_iq_{i+1}=\pm1$ so the pairs $(x_k,y_k)$ are relatively prime.