Let $\{ \cdot \}$ denote the fractional part function. Given $a, b, c, d \in \mathbb{R}$ such that $a \leqslant b$, $c \leqslant d$, $b - a = d - c$ and $\{ a x \} - \{ b x\} = \{c x \} - \{d x \}$ for $x \in \mathbb{Z}_{\geqslant 0}$, does it necessarily follow that there is a $\delta \in \mathbb{Z}$ such that $c = a + \delta$ and $d = b + \delta$? More generally, if $\{ a x \} - \{ b x\} = \{c x \} - \{d x \}$ for $x \in \mathbb{R}_{\geqslant 0}$, does it necessarily follow that $c = a$ and $d = b$?
I believe the answer is 'yes' in both cases, but I'm not quite sure how to prove it. If the constants are rational, then I believe a proof could follow by considering periodicity. Otherwise, if the constants are irrational, maybe a density argument would suffice.