Orbit of $(1/m, q)$ in torus where $q$ is an irrational number and $m \in \Bbb N$

Question

I am trying to understand the set $$\left\langle\left(\frac 1m, a\right)\right\rangle=\left\{\left(\frac km\pmod 1, ka\pmod 1\right): k \in \Bbb Z\right\}.$$ That is the orbit of $\left<(\frac 1m, a)\right>$ in $\Bbb T^2$ where $T^n((\frac 1m, a))=(\frac {n+1}m, (n+1)a)$ where $a$ is irrational. From my intuition it would be dense in each of the stripes in the picture. Can you help me prove this? Any step by step hint or entire proof or anything will be helpful. In the picture I used $m=10$.

Answer 1

Your intuition is correct. As a hint, each one of those $m$ vertical lines is invariant under $S=T^m$, and $S$ on each such line acts as an irrational rotation. Note that orbits of $S$ are included in orbits of $T$.