Title: Polynomial p-adic Low-Discrepancy Sequences Show Chinese title

Title: 多项式 p 进低差异序列

Abstract: The classic example of a low-discrepancy sequence in $\mathbb{Z}_p$ is $(x_n) = an+b$ with $a \in \mathbb{Z}_p^x$ and $b \in \mathbb{Z}_p$. Here we address the non-linear case and show that a polynomial $f$ generates a low-discrepancy sequence in $\mathbb{Z}_p$ if and only if it is a permutation polynomial $\mod p$ and $\mod p^2$. By this it is possible to construct non-linear examples of low-discrepancy sequences in $\mathbb{Z}_p$ for all primes $p$. Moreover, we prove a criterion which decides for any given polynomial in $\mathbb{Z}_p$ with $p \in \left\{ 3,5, 7\right\}$ if it generates a low-discrepancy sequence. We also discuss connections to the theories of Poissonian pair correlations and real discrepancy. Show Chinese abstract

Abstract: 在$\mathbb{Z}_p$中低差异序列的经典示例是$(x_n) = an+b$，其中包含$a \in \mathbb{Z}_p^x$和$b \in \mathbb{Z}_p$。 此处我们处理非线性情况，并证明多项式$f$在且仅在它是排列多项式$\mod p$且$\mod p^2$时，在$\mathbb{Z}_p$中生成低差异序列。 通过这种方法，可以为所有素数$p$构造$\mathbb{Z}_p$中的非线性低差异序列示例。 此外，我们证明了一个准则，该准则可以判断任何给定的多项式在$\mathbb{Z}_p$中带有$p \in \left\{ 3,5, 7\right\}$是否生成一个低偏差序列。 我们还讨论了与泊松对相关理论和实数偏差理论的联系。