As a result of recent studies on unidimensional low discrepancy sequences, we can assert that the original van der Corput sequences are the worst distributed with respect to various measures of irregularities of distribution among two large families of $(0,1)$–sequences, and even among all $(0,1)$–sequences for the star discrepancy $D^{*}$. We show in the present paper that it is not the case for the extreme discrepancy $D$ by producing two kinds of sequences which are the worst distributed among all $(0,1)$–sequences, with...

The second-named author recently suggested identifying the generating matrices of a digital (t,m,s)-net over the finite field ${}_q$ with an s × m matrix C over ${}_{q^m}$. More exactly, the entries of C are determined by interpreting the rows of the generating matrices as elements of ${}_{q^m}$. This paper introduces so-called Vandermonde nets, which correspond to Vandermonde-type matrices C, and discusses the quality parameter and the discrepancy of such nets. The methods that have been successfully used for the investigation...

For a prime p and an absolutely irreducible modulo p polynomial f(U,V) ∈ ℤ[U,V] we obtain an asymptotic formula for the number of solutions to the congruence f(x,y) ≡ a (mod p) in positive integers x ≤ X, y ≤ Y, with the additional condition gcd(x,y) = 1. Such solutions have a natural interpretation as solutions which are visible from the origin. These formulas are derived on average over a for a fixed prime p, and also on average over p for a fixed integer a.

Generalized van der Corput sequences are onedimensional, infinite sequences in the unit interval. They are generated from permutations in integer base $b$ and are the building blocks of the multi-dimensional Halton sequences. Motivated by recent progress of Atanassov on the uniform distribution behavior of Halton sequences, we study, among others, permutations of the form $P\left(i\right)=ai(modb)$ for coprime integers $a$ and $b$. We show that multipliers $a$ that either divide $b-1$ or $b+1$ generate van der Corput sequences with weak...