The Three Gap Theorem and Periodic Functions

TL;DR

This paper explores the extension of the Three Gap Theorem to various periodic functions, identifying classes where the theorem applies and others where it does not, revealing new insights into the combinatorics of fractional parts.

Contribution

The paper introduces a new problem extending the Three Gap Theorem to other periodic functions and proves results for specific classes, highlighting cases with unbounded gap lengths.

Findings

Analogous gap results hold for certain piecewise-linear periodic functions.

Some periodic functions exhibit no bound on the number of gap lengths.

The work broadens understanding of the combinatorics of fractional parts for periodic functions.

Abstract

The Three Gap Theorem, also known as the Steinhaus Conjecture, is a classical result on the combinatorics of the fractional part function, and has since been generalized in many ways. In this paper, we pose a new problem related to these results: for which other periodic functions does an analogue of the Three Gap Theorem hold? We prove analogous results for certain classes of piecewise-linear periodic functions and demonstrate the existence of functions for which no bound exists on the number of gap lengths.