Continuous Distribution Arising From the Three Gaps Theorem

TL;DR

This paper offers a new approach to the Three Gap Theorem, demonstrating that averaging over lpha results in a continuous distribution and providing bounds for the error terms.

Contribution

It introduces an alternative method to prove the continuous distribution arising from the Three Gap Theorem and offers explicit error bounds.

Findings

Averaging over lpha yields a continuous distribution.

The paper provides bounds for the approximation error.

The approach simplifies understanding the distribution of gaps.

Abstract

The well known Three Gap Theorem states that there are at most three gap sizes in the sequence of fractional parts $\{\alpha n\}_{n<N}$ . It is known that if one averages over {\alpha}, the distribution becomes continuous. We present an alternative approach, which establishes this averaged result and also provides good bounds for the error terms.