I wonder in which condition a bouncing ball can cover every point in the set $[0, 1]^2$.
Problem statement:
Given a straight line $l$ on $\mathbb{R}^2$. Let $$ C = \{(x - \lfloor x \rfloor, y - \lfloor y \rfloor) : (x, y) \in l\} $$
In which condition $C$ is dense on $[0, 1]^2$ ?
In which condition $C = [0, 1]^2$ ?
Some observations
Let $l$ be parameterized by $y = ax + b$ By symmetry, we can limit the domain as $a \in (0, \frac{1}{2})$, $b \in [0, 1]$, and possibly $x \in [0, \infty)$ instead of $\mathbb{R}$
- If $a$ is rational, $C \neq [0, 1]^2$
- If $a$ is irrational, $l$ goes through all vertical lines of the form $x = z$ for $z \in \mathbb{Z}$ at $y_z$. Then $\{y_z - \lfloor y_z \rfloor : z \in \mathbb{Z} \}$ is an infinite set, i.e. trajectory of the bouncing ball is infinitely long
Given Peano theorem, i.e the existence of space-filling curve. I am not sure how likely whether we have a definite answer to this.