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1
Content available remote

On existence and discrepancy of certain digital Niederreiter-Halton sequences

100%
Hofer R., Larcher G.
Acta Arithmetica
|
2010
|
tom 141
|
nr 4
369-394
2
Content available remote

Vandermonde nets

100%
Hofer R., Niederreiter H.
Acta Arithmetica
|
2014
|
tom 163
|
nr 2
145-160
EN
The second-named author recently suggested identifying the generating matrices of a digital (t,m,s)-net over the finite field $𝔽_{q}$ with an s × 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 ≤ 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 ≤ 2.
3
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Discrepancy estimates for some linear generalized monomials

100%
Hofer R., Ramaré O.
Acta Arithmetica
|
2016
|
tom 173
|
nr 2
183-196
EN
We consider sequences modulo one that are generated using a generalized polynomial over the real numbers. Such polynomials may also involve the integer part operation [·] additionally to addition and multiplication. A well studied example is the (nα) sequence defined by the monomial αx. Their most basic sister, $([nα]β)_{n≥0}$, is less investigated. So far only the uniform distribution modulo one of these sequences is resolved. Completely new, however, are the discrepancy results proved in this paper. We show in particular that if the pair (α,β) of real numbers is in a certain sense badly approximable, then the discrepancy satisfies a bound of order $𝓞_{α,β,ε}(N^{-1+ε})$.
4
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Distribution properties of sequences generated by Q-additive functions with respect to Cantor representation of integers

81%
Hofer R., Pillichshammer F., Pirsic G.
Acta Arithmetica
|
2009
|
tom 138
|
nr 2
179-200
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