A generalization of the 3d distance theorem

Abstract

Let P be a positive rational number. A function $f:\mathbb {R}\rightarrow \mathbb {R}$ has the finite gaps property mod P if the following holds: for any positive irrational $\alpha$ and positive integer M, when the values of $f(m\alpha )$, $1\le m\le M$, 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 the finite gaps property mod P. We also show that if f is the distance to the nearest integer function, then it has the finite gaps property mod 1 with $k_f\le 6$.