Article contents
A METRIC THEORY OF MINIMAL GAPS
Published online by Cambridge University Press: 21 May 2018
Abstract
We study the minimal gap statistic for fractional parts of sequences of the form ${\mathcal{A}}^{\unicode[STIX]{x1D6FC}}=\{\unicode[STIX]{x1D6FC}a(n)\}$, where ${\mathcal{A}}=\{a(n)\}$ is a sequence of distinct integers. Assuming that the additive energy of the sequence is close to its minimal possible value, we show that for almost all $\unicode[STIX]{x1D6FC}$, the minimal gap $\unicode[STIX]{x1D6FF}_{\min }^{\unicode[STIX]{x1D6FC}}(N)=\min \{\unicode[STIX]{x1D6FC}a(m)-\unicode[STIX]{x1D6FC}a(n)\hspace{0.2em}{\rm mod}\hspace{0.2em}1:1\leqslant m\neq n\leqslant N\}$ is close to that of a random sequence.
MSC classification
- Type
- Research Article
- Information
- Copyright
- Copyright © University College London 2018
References
- 3
- Cited by