Polynomial p-adic low-discrepancy sequences

Abstract

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