Extensions of Atanassov’s Methods for Halton Sequences

Comptes Rendus Mathematique, 2016

Let (H s (n)) n≥1 be an s-dimensional Halton's sequence. Let D N be the discrepancy of the sequence (H s (n)) N n=1. It is known that N D N = O (ln s N) as N → ∞. In this paper, we prove that this estimate is exact: lim N→∞ N ln −s (N)D N > 0.

1988

2019

Let ${\bf x}_0,{\bf x}_1,...$ be a sequence of points in $[0,1)^s$. A subset $S$ of $[0,1)^s$ is called a bounded remainder set if there exist two real numbers $a$ and $C$ such that, for every integer $N$, $$ | {\rm card}\{n <N \; | \; {\bf x}_{n} \in S\} - a N| <C . $$ Let $ ({\bf x}_n)_{n \geq 0} $ be an $s-$dimensional Halton-type sequence obtained from a global function field, $b \geq 2$, ${\bf \gamma} =(\gamma_1,...,\gamma_s)$, $\gamma_i \in [0, 1)$, with $b$-adic expansion $\gamma_i= \gamma_{i,1}b^{-1}+ \gamma_{i,2}b^{-2}+...$, $i=1,...,s$. In this paper, we prove that $[0,\gamma_1) \times ...\times [0,\gamma_s)$ is the bounded remainder set with respect to the sequence $({\bf x}_n)_{n \geq 0}$ if and only if \begin{equation} \nonumber \max_{1 \leq i \leq s} \max \{ j \geq 1 \; | \; \gamma_{i,j} \neq 0 \} < \infty. \end{equation} We also obtain the similar results for a generalized Niederreiter sequences, Xing-Niederreiter sequences and Niederreiter-Xing sequences.

2005

A generalization of interval-filling sequences given in [2] served as a powerful device for a representation of real numbers in canonical number systems. In this paper the coefficient set {0, 1, . . . , N} is replaced by an arbitrary finite set of real numbers. Though the complete characterization of interval-filling sequences of this type is given only in special cases, some general results are also obtained. Finally, an example shows that generally no necessary and sufficient condition of the “usual” form exists. Notation. Denote by P a fixed finite set of real numbers. Write P = {p0, p1, . . . , pN} where N ∈ N (i.e. N is a positive integer) and pi−1 < pi for i = 1, 2, . . . , N . Let Λ denote the set of real sequences λ = (λn) for which (i) |λn| > |λn+1| > 0 for every n ∈ N and

Proyecciones (Antofagasta)

In this article we introduce the notation difference operator ∆ m (m ≥ 0 be an integer) for studying some properties defined with interval numbers. We introduced the classes of sequence¯ (p)(∆ m),c(p)(∆ m) andc 0 (p)(∆ m) and investigate different algebraic properties like completeness, solidness, convergence free etc.

Journal of Number Theory, 1984

2020

Let (H(n))n≥0 be a 2−dimensional Halton’s sequence. Let D2((H(n)) N−1 n=0 ) be the L2-discrepancy of (Hn) N−1 n=0 . It is known that lim supN→∞(logN) D2(H(n)) N−1 n=0 > 0. In this paper, we prove that D2((H(n)) N−1 n=0 ) = O(logN) for N →∞, i.e., we found the smallest possible order of magnitude of L2-discrepancy of a 2-dimensional Halton’s sequence. The main tool is the theorem on linear forms in the p-adic logarithm.

The LS-sequences of points recently introduced by the author are a generalization of van der Corput sequences. They were constructed by reordering the points of the corresponding LS-sequences of partitions. Here we present another algorithm which coincides with the classical one for van der Corput sequences and is simpler to compute than the original construction. This algorithm is based on the representation of natural numbers in base L + S and gives the van der Corput sequence in base b if L = b and S = 0. In this construction, as well as in the van der Corput one, it is essential the inversion of digits of the representation in base L + S: in this paper we also give a nice geometrical explanation of this "magical" operation.

In this overview we show by examples, how to associate certain sequences in the higher-dimensional unit cube to suitable dynamical systems. We present methods and notions from ergodic theory that serve as tools for the study of low-discrepancy sequences and discuss an important technique, cuttingand-stacking of intervals.

Proyecciones (Antofagasta)

In this paper we introduce the sequence spaces c 0 i (∆) ,c i (∆) and i ∞ (∆) of interval numbers and study some of their algebraic and topological properties. Also we investigate some inclusion relations related to these spaces.