A metric theory of minimal gaps

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 $\delta_{\min}^\alpha(N)=\min\{\alpha a(m)-\alpha a(n)\bmod 1: 1\leq m\neq n\leq N\}$ is close to that of a random sequence.