A generalization of the 3d distance theorem

Let $P$ be a positive rational number. Call a function $f:\mathbb{R}\rightarrow\mathbb{R}$ to have $\textit{finite gaps property mod}$ $P$ if the following holds: for any positive irrational $\alpha$ and positive integer $M$, when the values of $f(m\alpha)$, $1\leq m\leq M$, are inserted mod $P$ into the interval $[0,P)$ and arranged in increasing order, the number of distinct gaps between successive terms is bounded by a constant $k_{f}$ which depends only on $f$. In this note, we prove a generalization of the 3d distance theorem of Chung and Graham. As a consequence, we show that a piecewise linear map with rational slopes and having only finitely many non-differentiable points has finite gaps property mod $P$. We also show that if $f$ is distance to the nearest integer function, then it has finite gaps property mod $1$ with $k_f\leq6$.