  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.
