Minimal Gaps and Additive Energy in real-valued sequences

Abstract

We study the minimal gap statistic for sequences of the form ${\left(\alpha x_n\right)}_{n=1}^{\infty }$ where ${\left(x_n\right)}_{n=1}^{\infty }$ is a sequence of real numbers, and its connection to the additive energy of ${\left(x_n\right)}_{n=1}^{\infty }$. Inspired by a recent paper of Aistleitner, El-Baz and Munsch we show conditionally on the Lindelöf Hypothesis that if the additive energy is of lowest possible order then for almost all $\alpha$, the minimal gap ${\delta }_{\min }^{\alpha }(N)=\min \left\{\alpha x_m-\alpha x_n\mathrm{mod}1:1\le m\ne n\le N\right\}$ is close to that of a random sequence, a result Rudnick showed for integer-valued sequences. We also show unconditional results in this direction, as well as some converse theorems about sequences with large additive energy.

28 pages. Submitted for publication