It's not difficult to prove that for $s\in\mathbb{R}\setminus\mathbb{Q}$ the set $S=\{n+ms\;|\;n\in\mathbb{Z},\;m\in\mathbb{N}\}$ is dense in $\mathbb{R}$.
When trying to solve this question, I come up with the following generalization: How uniformly is $S$ distributed as $m$ increases? Concretely:
Let $A\subseteq[0,1]$ be a measurable set. Let further $S_k=\{n+ms\;|\;n\in\mathbb{Z},\;m\in\{1,\ldots,k\}\}$ (so $S=\bigcup_k S_k$). Can we prove that $$\left|\frac{|S_k\cap A|}{|S_k\cap [0,1)\,|}-\mu(A)\right| \;\in\; O(f(k))$$ for some $f$ such that $\lim_{k\to\infty}f(k)=0$? (The faster $f$ approaches $0$ the better.)
The informal idea behind this claim is that the multiples of $s$ expressed as $S_k$ "fill" uniformly - whatever part $A\subseteq [0,1]$ we choose, it's being "filled" proportionally to the rest.
(Clearly this claim implies that $S$ is dense - for a given $r\in\mathbb{R}$ we pick $A=(r-\varepsilon,r+\varepsilon)$ and for some $k$ the set $A\cap S_k$ must be nonempty.)
This is the idea that formulated precisely in Weyl's criterion, a proof of which can be found in Stein's Fourier Analysis Volume 1. So it might be good reading for you.