A generalization of the 3d distance theorem

TL;DR

This paper generalizes the 3d distance theorem, establishing conditions under which functions exhibit bounded gap properties mod P, with applications to piecewise linear maps and the distance to the nearest integer function.

Contribution

It extends the 3d distance theorem to a broader class of functions and characterizes when they have finite gaps property mod P, including piecewise linear functions with rational slopes.

Findings

Piecewise linear maps with rational slopes have finite gaps property mod P.

Distance to the nearest integer function has finite gaps property with at most 6 gaps.

Generalization of the 3d distance theorem to new classes of functions.

Abstract

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 $k_f\leq6$.