实值序列的最小间隙与可加能量
Minimal Gaps and Additive Energy in real-valued sequences
摘要：
本文研究了$\left( αx_n \right)_{n = 1}^{\infty}$序列的最小间隙统计量，其中$\left( x_n \right)_{n = 1}^{\infty}$是实数序列，以及它与$\left( x_n \right)_{n = 1}^{\infty}$的加性能量的关系.受Aistleitner、El-Baz和Munsch最近的一篇论文的启发，我们有条件地证明了Lindelöf假设，如果加性能量是尽可能低的阶数，那么对于几乎所有的$α$，最小间隙$δ_{\min}^α (N) = \min \left\{ αx_m - αx_n \bmod \ 1 : 1 \leq m \neq n \leq N \right\}$接近于随机序列，这是Rudnick为整数值序列证明的结果。我们还显示在这个方向上的无条件的结果，以及一些逆定理序列具有大的附加能量。
英文摘要：
We study the minimal gap statistic for sequences of the form $\left( αx_n \right)_{n = 1}^{\infty}$ 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 $α$, the minimal gap $δ_{\min}^α (N) = \min \left\{ αx_m - αx_n \bmod \ 1 : 1 \leq m \neq n \leq N \right\}$ 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.
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