The Three Gap Theorem and Periodic Functions
Combinatorics
2022-02-15 v1 Number Theory
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.
Cite
@article{arxiv.2202.05921,
title = {The Three Gap Theorem and Periodic Functions},
author = {A. Suki Dasher and A. Hermida and Tian An Wong},
journal= {arXiv preprint arXiv:2202.05921},
year = {2022}
}