Polynomial p-adic Low-Discrepancy Sequences

TL;DR

This paper characterizes when polynomial functions generate low-discrepancy sequences in p-adic integers, providing criteria for certain primes and exploring connections to Poissonian pair correlations and real discrepancy.

Contribution

It extends the theory of low-discrepancy sequences to non-linear polynomials in p-adic integers and offers practical criteria for specific primes.

Findings

Polynomial sequences are low-discrepancy iff they are permutation polynomials mod p and p^2.

Constructs non-linear low-discrepancy sequences for all primes p.

Provides a criterion to identify low-discrepancy polynomials for p=3,5,7.

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.