Continuous distributions arising from the Three Gap Theorem

Geremías Polanco, Daniel Schultz, Alexandru Zaharescu

Research output: Contribution to journalArticlepeer-review


The well-known Three Gap Theorem states that there are at most three gap sizes in the sequence of fractional parts {αn}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.

Original languageEnglish (US)
Pages (from-to)1743-1764
Number of pages22
JournalInternational Journal of Number Theory
Issue number7
StatePublished - Nov 1 2016


  • Farey fractions
  • Three Gap Theorem
  • continuous distribution
  • uniform distribution

ASJC Scopus subject areas

  • Algebra and Number Theory


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