422,683 research outputs found
Dynamical consequences of a free interval: minimality, transitivity, mixing and topological entropy
We study dynamics of continuous maps on compact metrizable spaces containing
a free interval (i.e., an open subset homeomorphic to an open interval). A
special attention is paid to relationships between topological transitivity,
weak and strong topological mixing, dense periodicity and topological entropy
as well as to the topological structure of minimal sets. In particular, a
trichotomy for minimal sets and a dichotomy for transitive maps are proved.Comment: 21 page
Sampling-based proofs of almost-periodicity results and algorithmic applications
We give new combinatorial proofs of known almost-periodicity results for
sumsets of sets with small doubling in the spirit of Croot and Sisask, whose
almost-periodicity lemma has had far-reaching implications in additive
combinatorics. We provide an alternative (and L^p-norm free) point of view,
which allows for proofs to easily be converted to probabilistic algorithms that
decide membership in almost-periodic sumsets of dense subsets of F_2^n.
As an application, we give a new algorithmic version of the quasipolynomial
Bogolyubov-Ruzsa lemma recently proved by Sanders. Together with the results by
the last two authors, this implies an algorithmic version of the quadratic
Goldreich-Levin theorem in which the number of terms in the quadratic Fourier
decomposition of a given function is quasipolynomial in the error parameter,
compared with an exponential dependence previously proved by the authors. It
also improves the running time of the algorithm to have quasipolynomial
dependence instead of an exponential one.
We also give an application to the problem of finding large subspaces in
sumsets of dense sets. Green showed that the sumset of a dense subset of F_2^n
contains a large subspace. Using Fourier analytic methods, Sanders proved that
such a subspace must have dimension bounded below by a constant times the
density times n. We provide an alternative (and L^p norm-free) proof of a
comparable bound, which is analogous to a recent result of Croot, Laba and
Sisask in the integers.Comment: 28 page
Monads with arities and their associated theories
After a review of the concept of "monad with arities" we show that the
category of algebras for such a monad has a canonical dense generator. This is
used to extend the correspondence between finitary monads on sets and Lawvere's
algebraic theories to a general correspondence between monads and theories for
a given category with arities. As application we determine arities for the free
groupoid monad on involutive graphs and recover the symmetric simplicial nerve
characterisation of groupoids.Comment: New introduction; Section 1 shortened and redispatched with Section
2; Subsections on symmetric operads (3.14) and symmetric simplicial sets
(4.17) added; Bibliography complete
Open sets satisfying systems of congruences
A famous result of Hausdorff states that a sphere with countably many points
removed can be partitioned into three pieces A,B,C such that A is congruent to
B (i.e., there is an isometry of the sphere which sends A to B), B is congruent
to C, and A is congruent to (B union C); this result was the precursor of the
Banach-Tarski paradox. Later, R. Robinson characterized the systems of
congruences like this which could be realized by partitions of the (entire)
sphere with rotations witnessing the congruences. The pieces involved were
nonmeasurable. In the present paper, we consider the problem of which systems
of congruences can be satisfied using open subsets of the sphere (or related
spaces); of course, these open sets cannot form a partition of the sphere, but
they can be required to cover "most of" the sphere in the sense that their
union is dense. Various versions of the problem arise, depending on whether one
uses all isometries of the sphere or restricts oneself to a free group of
rotations (the latter version generalizes to many other suitable spaces), or
whether one omits the requirement that the open sets have dense union, and so
on. While some cases of these problems are solved by simple geometrical
dissections, others involve complicated iterative constructions and/or results
from the theory of free groups. Many interesting questions remain open.Comment: 44 page
Infinite Lineability: On the Abundance of Dense Subspaces
In this paper, we investigate the concept of infinite dense-lineability
recently introduced by M. Calder\'on-Moreno, P. Gerlach-Mena and J.
Prado-Bassas. We answer a question posed by the authors about the equivalence
between infinite (pointwise) dense-lineability and (pointwise)
dense-lineability. We prove that the equivalence always holds in
first-countable topological vector spaces and under some assumptions about the
weight of the topology. However, the equivalence is not always true, as shown
in an example. Furthermore, we introduce the notions of infinite
-dense-lineability and infinite (strongly) dense-algebrability
and obtain some analogous results in these cases. We also obtain a criterion
for strongly dense-algebrability for sets of the form , where
is a free algebra and is a free subalgebra of .Comment: 12 page
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