Continuous Distribution Arising From the Three Gaps Theorem

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 α, the distribution becomes continuous. We present an alternative approach, which establishes this averaged result and also provides good bounds for the error terms.

23 pages including references, Four figures