# Johannes Kepler Symposium für Mathematik

Im Rahmen des Johannes-Kepler-Symposiums für Mathematik wird Assist.-Prof. Mag. Dr. Roswitha Hofer, Institute of Financial Mathematics and Applied Number Theory, JKU Linz, am Wed, April 27, 2016 um 17:15 Uhr im HS 13 einen öffentlichen Vortrag (mit anschließender Diskussion) zum Thema "Construction of Low-Discrepancy Point Sets and Sequences, and the Distribution of Subsequences" halten, zu dem die Veranstalter des Symposiums,

O.Univ.-Prof. Dr. Ulrich Langer,
Univ.-Prof. Dr. Gerhard Larcher
A.Univ.-Prof. Dr. Jürgen Maaß, und
die ÖMG (Österreichische Mathematische Gesellschaft)

Series B - Mathematical Colloquium:

The intention is to present new mathematical results for an audience interested in general mathematics.

### Construction of Low-Discrepancy Point Sets and Sequences, and the Distribution of Subsequences

The first part of the talk outlines ten research articles which were developed with several coauthors within the last five years. These papers are listed below and sorted into three groups, [C1]–[C5], [V1]–[V2], and [S1]–[S3]. Their thematic frame is uniform distribution theory and its application to quasi-Monte Carlo integration. These two topics have a strong history in Austria evidenced for instance by the work of Edmund Hlawka and of Harald Niederreiter, two most distinguished Austrian mathematicians of worldwide renown. Uniform distribution theory and quasi-Monte Carlo methods are still actively studied by Austrian mathematicians these days, note for instance the Special Research Program Quasi-Monte Carlo Methods: Theory and Application funded by the FWF.

The celebrated Koksma-Hlawka inequality,

$\left|\int_{[0,1]^s} f({\mathbf x})d{\mathbf x}-\frac{1}{N}\sum_{n=0}^{N-1}f({\mathbf x}_n)\right| \leq V(f)D_N^*({\mathbf x}_0,{\mathbf x}_1,\dots,{\mathbf x}_{N-1})$,

links quasi-Monte Carlo integration with uniform distribution theory, as the star discrepancy

$D_N^*({\mathbf x}_0,{\mathbf x}_1,\dots,{\mathbf x}_{N-1}):=\sup_{{\mathbf a}\in[0,1]^s}\left|\frac{\#\{n:0\leq n\lt N, \,{\mathbf x}_n\in[{\mathbf 0},{\mathbf a})\}}{N}-\lambda_s([{\mathbf 0},{\mathbf a}))\right|$

is a measure of the uniformity of the point set $\{{\mathbf x}_0,{\mathbf x}_1,\dots,{\mathbf x}_{N-1}\}$ in $[0,1]^s$ that is used by the quadrature rule. A great part of the numerous research articles of Harald Niederreiter provides construction methods for point sets and sequences having small star discrepancy. The research articles [C1]-[C5] and [V1]-[V2], partially joint work with Harald Niederreiter, are devoted to establishing new constructions of sequences and point sets having low star discrepancy. The research articles [S1]-[S3] are dedicated to the study of the uniform distribution of special subsequences. The distribution of subsequences is very actively studied. Note for instance the celebrated work of Rivat and Mauduit on the Gelfond conjectures which address the distribution of the sum of digits function over squares and over prime numbers.

The second and major part of the talk singles out the five research articles [C1]-[C5] and concentrates on relevant previous and recent work before giving some selected details of their content.

[C1] Roswitha Hofer and Gottlieb Pirsic.
An explicit construction of finite-row digital (0; s)-sequences.
Unif. Distrib. Theory, 6(2):13–30, 2011.
[C2] Roswitha Hofer.
A construction of digital (0; s)-sequences involving finite-row generator matrices.
Finite Fields Appl., 18(3):587–596, 2012.
[C3] Roswitha Hofer.
A construction of low-discrepancy sequences involving finite-row digital (t; s)-sequences.
Monatsh. Math., 171(1):77–89, 2013.
[C4] Roswitha Hofer and Harald Niederreiter.
A construction of (t; s)-sequences with finite-row generating matrices using global function fields.
Finite Fields Appl., 21:97–110, 2013.
[C5] Roswitha Hofer.
Generalized Hofer-Niederreiter sequences and their discrepancy from an (U; e; s)-point of view.
J. Complexity, 31(2):260–276, 2015.
[V1] Roswitha Hofer and Harald Niederreiter.
Vandermonde nets.
Acta Arith., 163(2):145–160, 2014.
[V2] Roswitha Hofer and Harald Niederreiter.
Vandermonde nets and Vandermonde sequences.
Monte Carlo and Quasi-Monte Carlo Methods 2014. (R. Cools and D. Nuyens eds.), Springer, Berlin Heidelberg New York, to appear., 2016.
[S1] Roswitha Hofer, Gerhard Larcher, and Heidrun Zellinger.
On the digits of squares and the distribution of quadratic subsequences of digital sequences.
Proc. Am. Math. Soc., 141(5):1551–1565, 2013.
[S2] Roswitha Hofer and Heidrun Zellinger.
Distribution properties of certain subsequences of digital sequences and their hybrid version.
Unif. Distrib. Theory, 8(2):121–140, 2013.
[S3] Roswitha Hofer and Olivier Ramaré.
Discrepancy estimates for some linear generalized monomials.
Acta Arith. Online first. DOI: 10.4064/aa8164-12-2015.