Minimal Gaps and Additive Energy in real-valued sequences

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 Lindel\"{o}f Hypothesis that if the additive energy is of lowest possible order then for almost all $\alpha$, the minimal gap $\delta_{\min}^{\alpha} (N) = \min \left\{ \alpha x_m - \alpha x_n \bmod \ 1 : 1 \leq m \neq n \leq N \right\}$ 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.