Polynomial p-adic Low-Discrepancy Sequences

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 $\mathrm{mod}p$ and $\mathrm{mod}{p}^2$. 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.