538 research outputs found
Quasi-selective ultrafilters and asymptotic numerosities
We isolate a new class of ultrafilters on N, called "quasi-selective" because
they are intermediate between selective ultrafilters and P-points. (Under the
Continuum Hypothesis these three classes are distinct.) The existence of
quasi-selective ultrafilters is equivalent to the existence of "asymptotic
numerosities" for all sets of tuples of natural numbers. Such numerosities are
hypernatural numbers that generalize finite cardinalities to countable point
sets. Most notably, they maintain the structure of ordered semiring, and, in a
precise sense, they allow for a natural extension of asymptotic density to all
sequences of tuples of natural numbers.Comment: 27 page
Central sets and substitutive dynamical systems
In this paper we establish a new connection between central sets and the
strong coincidence conjecture for fixed points of irreducible primitive
substitutions of Pisot type. Central sets, first introduced by Furstenberg
using notions from topological dynamics, constitute a special class of subsets
of \nats possessing strong combinatorial properties: Each central set
contains arbitrarily long arithmetic progressions, and solutions to all
partition regular systems of homogeneous linear equations. We give an
equivalent reformulation of the strong coincidence condition in terms of
central sets and minimal idempotent ultrafilters in the Stone-\v{C}ech
compactification \beta \nats . This provides a new arithmetical approach to
an outstanding conjecture in tiling theory, the Pisot substitution conjecture.
The results in this paper rely on interactions between different areas of
mathematics, some of which had not previously been directly linked: They
include the general theory of combinatorics on words, abstract numeration
systems, tilings, topological dynamics and the algebraic/topological properties
of Stone-\v{C}ech compactification of \nats.Comment: arXiv admin note: substantial text overlap with arXiv:1110.4225,
arXiv:1301.511
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
Ultrafilters on -spaces
For a discrete group and a discrete -space , we identify the
Stone-\v{C}ech compactifications and with the sets of all
ultrafilters on and , and apply the natural action of on
to characterize large, thick, thin, sparse and scattered subsets of
. We use -invariant partitions and colorings to define -selective and
-Ramsey ultrafilters on . We show that, in contrast to the
set-theoretical case, these two classes of ultrafilters are distinct. We
consider also universally thin ultrafilters on , the -points, and
study interrelations between these ultrafilters and some classical ultrafilters
on
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