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Mordechay B. Levin has constructed a number $\lambda$ which is normal in base 2, and such that the sequence $(\left\{2^n \lambda\right\})_{n=0,1,2,\ldots}$ has very small discrepancy $D_N$. Indeed we have $N\cdot D_N = \mathcal{O} \left(\left(\log N\right)^2\right)$. This construction technique of Levin was generalized by Becher and Carton, who generated normal numbers via perfect nested necklaces, and they showed that for these normal numbers the same upper discrepancy estimate holds as for the special example of Levin. In this paper now we derive an upper discrepancy bound for so-called semi-perfect nested necklaces and show that for the Levin's normal number in arbitrary prime base $p$ this upper bound for the discrepancy is best possible, i.e., $N\cdot D_N \geq c\left(\log N\right)^2$ with $c>0$ for infinitely many $N$. This result generalizes a previous result where we ensured for the special example of Levin for the base $p=2$, that $N\cdot D_N =O( \left(\log N\right)^2)$ is best possible in $N$. So far it is known by a celebrated result of Schmidt that for any sequence in $[0,1)$, $N\cdot D_N\geq c \log N$ with $c>0$ for infinitely many $N$. So there is a gap of a $\log N$ factor in the question, what is the best order for the discrepancy in $N$ that can be achieved for a normal number. Our result for Levin's normal number in any prime base on the one hand might support the guess that $O( \left(\log N\right)^2)$ is the best order in $N$ that can be achieved by a normal number, while generalizing the class of known normal numbers by introducing e.g. semi-perfect necklaces on the other hand might help for the search of normal numbers that satisfy smaller discrepancy bounds in $N$ than $N\cdot D_N=O( \left(\log N\right)^2)$., Comment: 29 pages

Mordechay Levin has constructed a number $\alpha$ which is normal in base 2, and such that the sequence $\left\{2^n \alpha\right\}_{n=0,1,2,\ldots}$ has very small discrepancy $D_N$. Indeed we have $N\cdot D_N = \mathcal{O} \left(\left(\log N\right)^2\right)$. That means, that $\alpha$ is normal of extremely high quality. In this paper we show that this estimate is best possible, i.e., $N\cdot D_N \geq c \cdot \left(\log N\right)^2$ for infinitely many $N$., Comment: There was a gap in the proof in a previous version. Here now is the corrected version

The focus of this paper is on formal power series analogs of the golden ratio. We are interested in both their continued fractions expansions as well as their Laurent series expansions. Our approach studies the Hankel matrices that are determined using the coefficients of the Laurent series expansions., Comment: 26 pages, 4 figures

Niederreiter and Halton sequences are two prominent classes of multi-dimensional sequences which are widely used in practice for numerical integration methods because of their excellent distribution qualities. In this paper, we show that these sequences - even though they are uniformly distributed - fail to satisfy the stronger property of Poissonian pair correlations. This extends already established results for one-dimensional sequences and confirms a conjecture of Larcher and Stockinger. The proofs rely on a general tool which identifies specific regularities of a sequence to be sufficient for not having Poissonian pair correlations., Comment: 18 pages, generalized statement of Theorem 1 from Faure to Niederreiter sequences

In this paper we extend a result of Levin, who proved a lower bound on the star discrepancy of generalized Halton sequences in positive integer bases, to rational bases.

In this paper we consider finite hybrid point sets that are the digital analogs to finite hybrid point sets introduced by Kritzer. Kritzer considered hybrid point sets that are a combination of lattice point sets and Hammersley point sets constructed using the ring of integers and the field of rational numbers. In this paper we consider finite hybrid point sets whose components stem from Halton-type Hammersley Point sets and lattice point sets which are constructed using the arithmetic of the ring of polynomials and the field of rational functions over a finite field. We present existence results for such finite hybrid point sets with low discrepancy., Comment: 16 pages

We give a characterization of all matrices $A,B,C \in \mathbb{F}_{2}^{m \times m}$ which generate a $(0,m,3)$-net in base $2$ and a characterization of all matrices $B,C\in\mathbb{F}_{2}^{\mathbb{N}\times\mathbb{N}}$ which generate a $(0,2)$-sequence in base $2$., Comment: 7 pages

In this paper we investigate the distribution properties of hybrid sequences which are made by combining Halton sequences in the ring of polynomials and digital Kronecker sequences. We give a full criterion for the uniform distribution and prove results on the discrepancy of such hybrid sequences., Comment: 23 pages

In this paper we investigate the special subsequence of the Halton sequence indexed by $\lfloor\beta n\rfloor$ with $\beta \in \mathbb{R}$ and prove a metric almost low- discrepancy result., Comment: 8 pages, 1 figure

We introduce a hybridization of digital sequences with uniformly distributed sequences in the domain of $b$-adic integers, $\mathbb Z_{b}, b\in\mathbb N\setminus\{1\}$, by using such sequences as input for generating matrices. The generating matrices are then naturally required to have finite row-lengths. We exhibit some relations of the `classical' digital method to our extended version, and also give several examples of new constructions with their respective quality assessments in terms of $t,\mathbf T$ and discrepancy., Comment: 17 pages, 2 figures

We consider the star discrepancy of two-dimensional sequences made up as a hybrid between a Kronecker sequence and a perturbed Halton sequence in base 2, where the perturbation is achieved by a digital-sequence construction in the sense of Niederreiter whose generating matrix contains a periodic perturbing sequence of a given period length. Under the assumption that the Kronecker sequence involves a parameter with bounded continued fraction coefficients sharp discrepancy estimates are obtained. Furthermore, we study the problem from a metric point of view as well. Finally, we also present sharp general and tight metric bounds for certain lacunary trigonometric products which appear to be strongly related to these problems., Comment: 28 pages, 2 figures. We are thankful for the helpful comments of E. Fouvry, as well as for the comments of the anonymous referee, which are considered in this resubmission

One of the fundamental theorems of uniform distribution theory states that the fractional parts of the sequence $(n \alpha)_{n \geq 1}$ are uniformly distributed modulo one (u.d. mod 1) for every irrational number $\alpha$. Another important result of Weyl states that for every sequence $(n_k)_{k \geq 1}$ of distinct positive integers the sequence of fractional parts of $(n_k \alpha)_{k \geq 1}$ is u.d. mod 1 for almost all $\alpha$. However, in this general case it is usually extremely difficult to classify those $\alpha$ for which uniform distribution occurs, and to measure the speed of convergence of the empirical distribution of $(\{n_1 \alpha\}, ..., \{n_N \alpha\})$ towards the uniform distribution. In the present paper we investigate this problem in the case when $(n_k)_{k \geq 1}$ is the Thue--Morse sequence of integers, which means the sequence of positive integers having an even sum of digits in base 2. In particular we utilize a connection with lacunary trigonometric products $\prod^{L}_{\ell=0} |\sin \pi 2^{\ell} \alpha |$, and by giving sharp metric estimates for such products we derive sharp metric estimates for exponential sums of $(n_{k} \alpha)_{k \geq 1}$ and for the discrepancy of $(\{n_{k} \alpha\})_{k \geq 1}.$ Furthermore, we comment on the connection between our results and an open problem in the metric theory of Diophantine approximation, and we provide some explicit examples of numbers $\alpha$ for which we can give estimates for the discrepancy of $(\{n_{k} \alpha\})_{k \geq1}$., Comment: 46 pages. Version 2: fixed typo in statement of Theorem 2

The authors recently introduced so-called Vandermonde nets. These digital nets share properties with the well-known polynomial lattices. For example, both can be constructed via component-by-component search algorithms. A striking characteristic of the Vandermonde nets is that for fixed $m$ an explicit construction of $m \times m$ generating matrices over the finite field $F_q$ is known for dimensions $s \le q+1$. This paper extends this explicit construction in two directions. We give a maximal extension in terms of $m$ by introducing a construction algorithm for $\infty \times \infty$ generating matrices for digital sequences over $F_q$, which works in the rational function field over $F_q$. Furthermore, we generalize this method to global function fields of positive genus, which leads to extensions in the dimension $s$.

The second author recently suggested to identify the generating matrices of a digital $(t,m,s)$-net over the finite field $F_q$ with an $s \times m$ matrix $C$ over $F_{q^m}$. More exactly, the entries of $C$ are determined by interpreting the rows of the generating matrices as elements of $F_{q^m}$. 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 of polynomial lattice point sets and hyperplane nets are applied to this new class of digital nets. In this way, existence results for small quality parameters and good discrepancy bounds are obtained. Furthermore, a first step towards component-by-component constructions is made. A novelty of this new class of nets is that explicit constructions of Vandermonde nets over $F_q$ in dimensions $s \le q+1$ with best possible quality parameter can be given. So far, good explicit constructions of the competing polynomial lattice point sets are known only in dimensions $s \le 2$.

For any prime power $q$ and any dimension $s \ge 1$, we present a construction of $(t,s)$-sequences in base $q$ with finite-row generating matrices such that, for fixed $q$, the quality parameter $t$ is asymptotically optimal as a function of $s$ as $s \to \infty$. This is the first construction of $(t,s)$-sequences that yields finite-row generating matrices and asymptotically optimal quality parameters at the same time. The construction is based on global function fields. We put the construction into the framework of $(u,{\bf e},s)$-sequences that was recently introduced by Tezuka. In this way we obtain in many cases better discrepancy bounds for the constructed sequences than by previous methods for bounding the discrepancy.