Continuous distributions arising from the three gap theorem. (English) Zbl 1396.11041
Summary: 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.
MSC:
| 11B57 | Farey sequences; the sequences \(1^k, 2^k, \dots\) |
| 11K36 | Well-distributed sequences and other variations |
| 11J71 | Distribution modulo one |
References:
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