论文标题
关于$ n^α$ mod 1的相关性
On the correlations of $n^α$ mod 1
论文作者
论文摘要
均匀分布模量理论的一个众所周知的结果（可以追溯到fejér和csillag）指出，序列$（n^α）_ {n^α）_ {n \ ge1} $的分数零件$ \ {n^α\} $在$α> 0 $的情况下是单位间隔均匀分布的。为了将这些知识提高到本地统计信息，$ k $级别的相关函数$（\ {n^α\}）_ {n \ geq1} $至关重要。我们证明，对于每个$ k \ ge2，$ k $级别的相关函数$ r_k $是poissonian，几乎每$α> 4K^2-4K-1 $。
A well known result in the theory of uniform distribution modulo one (which goes back to Fejér and Csillag) states that the fractional parts $\{n^α\}$ of the sequence $(n^α)_{n\ge1}$ are uniformly distributed in the unit interval whenever $α>0$ is not an integer. For sharpening this knowledge to local statistics, the $k$-level correlation functions of the sequence $(\{n^α\})_{n\geq1}$ are of fundamental importance. We prove that for each $k\ge2,$ the $k$-level correlation function $R_k$ is Poissonian for almost every $α>4k^2-4k-1$.
