5,089 research outputs found
Intersections of multiplicative translates of 3-adic Cantor sets
Motivated by a question of Erd\H{o}s, this paper considers questions
concerning the discrete dynamical system on the 3-adic integers given by
multiplication by 2. Let the 3-adic Cantor set consist of all 3-adic integers
whose expansions use only the digits 0 and 1. The exception set is the set of
3-adic integers whose forward orbits under this action intersects the 3-adic
Cantor set infinitely many times. It has been shown that this set has Hausdorff
dimension 0. Approaches to upper bounds on the Hausdorff dimensions of these
sets leads to study of intersections of multiplicative translates of Cantor
sets by powers of 2. More generally, this paper studies the structure of finite
intersections of general multiplicative translates of the 3-adic Cantor set by
integers 1 < M_1 < M_2 < ...< M_n. These sets are describable as sets of 3-adic
integers whose 3-adic expansions have one-sided symbolic dynamics given by a
finite automaton. As a consequence, the Hausdorff dimension of such a set is
always of the form log(\beta) for an algebraic integer \beta. This paper gives
a method to determine the automaton for given data (M_1, ..., M_n).
Experimental results indicate that the Hausdorff dimension of such sets depends
in a very complicated way on the integers M_1,...,M_n.Comment: v1, 31 pages, 6 figure
L_p- and S_{p,q}^rB-discrepancy of (order 2) digital nets
Dick proved that all order digital nets satisfy optimal upper bounds of
the -discrepancy. We give an alternative proof for this fact using Haar
bases. Furthermore, we prove that all digital nets satisfy optimal upper bounds
of the -discrepancy for a certain parameter range and enlarge that
range for order digitals nets. -, - and -discrepancy is considered as well
Explicit constructions of point sets and sequences with low discrepancy
In this article we survey recent results on the explicit construction of
finite point sets and infinite sequences with optimal order of
discrepancy. In 1954 Roth proved a lower bound for the
discrepancy of finite point sets in the unit cube of arbitrary dimension. Later
various authors extended Roth's result to lower bounds also for the
discrepancy and for infinite sequences. While it was known
already from the early 1980s on that Roth's lower bound is best possible in the
order of magnitude, it was a longstanding open question to find explicit
constructions of point sets and sequences with optimal order of
discrepancy. This problem was solved by Chen and Skriganov in 2002 for finite
point sets and recently by the authors of this article for infinite sequences.
These constructions can also be extended to give optimal order of the
discrepancy of finite point sets for . The
main aim of this article is to give an overview of these constructions and
related results
From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules
In 1935 J.G. van der Corput introduced a sequence which has excellent uniform
distribution properties modulo 1. This sequence is based on a very simple
digital construction scheme with respect to the binary digit expansion.
Nowadays the van der Corput sequence, as it was named later, is the prototype
of many uniformly distributed sequences, also in the multi-dimensional case.
Such sequences are required as sample nodes in quasi-Monte Carlo algorithms,
which are deterministic variants of Monte Carlo rules for numerical
integration. Since its introduction many people have studied the van der Corput
sequence and generalizations thereof. This led to a huge number of results.
On the occasion of the 125th birthday of J.G. van der Corput we survey many
interesting results on van der Corput sequences and their generalizations. In
this way we move from van der Corput's ideas to the most modern constructions
of sequences for quasi-Monte Carlo rules, such as, e.g., generalized Halton
sequences or Niederreiter's -sequences
Rigid abelian groups and the probabilistic method
The construction of torsion-free abelian groups with prescribed endomorphism
rings starting with Corner's seminal work is a well-studied subject in the
theory of abelian groups. Usually these construction work by adding elements
from a (topological) completion in order to get rid of (kill) unwanted
homomorphisms. The critical part is to actually prove that every unwanted
homomorphism can be killed by adding a suitable element. We will demonstrate
that some of those constructions can be significantly simplified by choosing
the elements at random. As a result, the endomorphism ring will be almost
surely prescribed, i.e., with probability one.Comment: 12 pages, submitted to the special volume of Contemporary Mathematics
for the proceedings of the conference Group and Model Theory, 201
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