Polynomial p-adic Low-Discrepancy Sequences

The classic example of a low-discrepancy sequence in Zₚ is (xₙ) = an+b with a ∈ Zₚˣ and b ∈ Zₚ. Here we address the non-linear case and show that a polynomial f generates a low-discrepancy sequence in Zₚ if and only if it is a permutation polynomial p and p². By this it is possible to construct non-linear examples of low-discrepancy sequences in Zₚ for all primes p. Moreover, we prove a criterion which decides for any given polynomial in Zₚ with p ∈ \ 3,5, 7\ if it generates a low-discrepancy sequence. We also discuss connections to the theories of Poissonian pair correlations and real discrepancy.