arXiv subjects
Publications and source records attributed to Gerhard Larcher.
It is known that there is a constant $c>0$ such that for every sequence $x_1, x_2,\ldots$ in $[0,1)$ we have for the star discrepancy $D^{*}_N$ of the first $N$ elements of the sequence that $N D^{*}_N\geq c\cdot \log N$ holds for infinitely many $N$. Let $c^{*}$ be the supremum of all such $c$ with this property. We show $c^{*}>0.065664679\ldots$, thereby slightly improving the estimates known until now.
An important result of H. Weyl states that for every sequence $\left(a_{n}\right)_{n\geq 1}$ of distinct positive integers the sequence of fractional parts of $\left(a_{n} α\right)_{n \geq1}$ is uniformly distributed modulo one for almost all $α$. However, in general it is a very hard problem to calculate the precise order of convergence of the discrepancy $D_{N}$ of $\left(\left\{a_{n} α\right\}\right)_{n \geq 1}$ for almost all $α$. By a result of R. C. Baker this discrepancy always satisfies $N D_{N} = \mathcal{O} \left(N^{\frac{1}{2}+\varepsilon}\right)$ for almost all $α$ and all $\varepsilon >0$. In the present note for arbitrary $γ\in \left(0, \frac{1}{2}\right]$ we construct a sequence $\left(a_{n}\right)_{n \geq 1}$ such that for almost all $α$ we have $ND_{N} = \mathcal{O} \left(N^γ\right)$ and $ND_{N} = Ω\left(N^{γ-\varepsilon}\right)$ for all $\varepsilon > 0$, thereby proving that any prescribed metric discrepancy behavior within the admissible range can actually be realized.
One of the fundamental theorems of uniform distribution theory states that the fractional parts of the sequence $(n α)_{n \geq 1}$ are uniformly distributed modulo one (u.d. mod 1) for every irrational number $α$. 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 $(n_k α)_{k \geq 1}$ is u.d. mod 1 for almost all $α$. However, in this general case it is usually extremely difficult to classify those $α$ for which uniform distribution occurs, and to measure the speed of convergence of the empirical distribution of $(\{n_1 α\}, ..., \{n_N α\})$ 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 $\prod^{L}_{\ell=0} |\sin π2^{\ell} α|$, and by giving sharp metric estimates for such products we derive sharp metric estimates for exponential sums of $(n_{k} α)_{k \geq 1}$ and for the discrepancy of $(\{n_{k} α\})_{k \geq 1}.$ 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 $α$ for which we can give estimates for the discrepancy of $(\{n_{k} α\})_{k \geq1}$.
In this paper we show discrepancy bounds for index-transformed uniformly distributed sequences. From a general result we deduce very tight lower and upper bounds on the discrepancy of index-transformed van der Corput-, Halton-, and $(t,s)$-sequences indexed by the sum-of-digits function. We also analyze the discrepancy of sequences indexed by other functions, such as, e.g., $\lfloor n^α\rfloor$ with $0 < α< 1$.
In this paper we give a short biography of Harald Niederreiter and we spotlight some cornerstones from his wide-ranging work. We focus on his results on uniform distribution, algebraic curves, polynomials and quasi-Monte Carlo methods. In the flavor of Harald's work we also mention some applications including numerical integration, coding theory and cryptography.
In this paper we give an overview of recent results on (upper and lower) discrepancy estimates for (concrete) sequences in the unit-cube. In particular we also give an overview of discrepancy estimates for certain classes of hybrid sequences. Here by a hybrid sequence we understand an $(s+t)$-dimensional sequence which is a combination of an $s$-dimensional sequence of a certain type (e.g. Kronecker-, Niederreiter-, Halton-,... type) and a $t$-dimensional sequence of another type. The analysis of the discrepancy of hybrid sequences (and of their components) is a rather current and vivid branch of research. We give a collection of some challenging open problems on this topic.
It is known that there is a constant $c > 0$ such that for every sequence $x_1, x_2, \ldots$ in $[0,1)$ we have for the star discrepancy $D_N^*$ of the first $N$ elements of the sequence that $N D_N^* \ge c \cdot \log N$ holds for infinitely many $N$. Let $c^*$ be the supremum of all such $c$ with this property. We show $c^* > 0.0646363$, thereby improving the until now known estimates.
In this paper we analyze and compare the use of Monte Carlo, Quasi-Monte Carlo and hybrid Monte Carlo-methods in the credit risk management system Credit Metrics by J.P.Morgan. We show that hybrid sequences used for simulations, in a suitable way, in many relevant situations, perform better than pure Monte Carlo and pure Quasi-Monte Carlo methods, and they essentially never perform worse than these methods.
Digital Kronecker-sequences are a non-archimedean analog of classical Kronecker-sequences whose construction is based on Laurent series over a finite field. In this paper it is shown that for almost all digital Kronecker-sequences the star discrepancy satisfies $D_N^\ast \ge c(q,s) (\log N)^s \log \log N$ for infinitely many $N \in \NN$, where $c(q,s)>0$ only depends on the dimension $s$ and on the order $q$ of the underlying finite field, but not on $N$. This result shows that a corresponding metrical upper bound due to Larcher is up to some $\log \log N$ term best possible.
In this paper we study uniform distribution properties of digital sequences over a finite field of prime order. In 1998 it was shown by Larcher that for almost all $s$-dimensional digital sequences the star discrepancy $D_N^\ast$ satisfies an upper bound of the form $D_N^\ast=O((\log N)^s (\log \log N)^{2+\varepsilon})$ for any $\varepsilon>0$. Generally speaking it is much more difficult to obtain good lower bounds for specific sequences than upper bounds. Here we show that Larchers result is best possible up to some $\log \log N$ term. More detailed, we prove that for almost all $s$-dimensional digital sequences the star discrepancy satisfies $D_N^\ast \ge c(q,s) (\log N)^s \log \log N$ for infinitely many $N \in \NN$, where $c(q,s)>0$ only depends on $q$ and $s$ but not on $N$.