A metric theory of minimal gaps

Abstract

We study the minimal gap statistic for fractional parts of sequences of the form ${\mathcal{A}}^{\alpha }=\{\alpha a(n)\}$ where $\mathcal{A}=\{a(n)\}$ is a sequence of distinct of integers. Assuming that the additive energy of the sequence is close to its minimal possible value, we show that for almost all $\alpha$, the minimal gap ${\delta }_{\min }^{\alpha }(N)=\min \{\alpha a(m)-\alpha a(n)\mathrm{mod}1:1\le m\ne n\le N\}$ is close to that of a random sequence.

Version 2: Fixed a small bug pointed out by Niclas Technau in the statement of section 3, and added references to few gaps in sequences of fractional parts suggested by Andrew Granville

References in corpus (1)

• Additive energy and the metric Poissonian property

Cited by in corpus (2)

• A Jacobi spectral method for computing eigenvalue gaps and their distribution statistics of the fractional Schrödinger operator
• Decoupling theorems for the Duffin-Schaeffer problem