Polynomial p-adic low-discrepancy sequences

The classic example of a low-discrepancy sequence in Z p is ( x n ) = a n + b with a ∈ Z p × and b ∈ 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 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 ∈ { 3 , 5 , 7 } if it generates a low-discrepancy sequence. We also discuss connections to the theories of Poissonian pair correlations and real discrepancy.