Minimal gaps and additive energy in real-valued sequences

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