arXiv subjects

Will Brian

Publications and source records attributed to Will Brian.

32 records · Page 2Linked to original sources

arXiv2019

Choosing between incompatible ideals

Suppose $\mathcal I$ and $\mathcal J$ are proper ideals on some set $X$. We say that $\mathcal I$ and $\mathcal J$ are incompatible if $\mathcal I \cup \mathcal J$ does not generate a proper ideal. Equivalently, $\mathcal I$ and $\mathcal J$ are incompatible if there is some $A \subseteq X$ such that $A \in \mathcal I$ and $X \setminus A \in \mathcal J$. If some $B \subseteq X$ is either in $\mathcal I \setminus \mathcal J$ or in $\mathcal J \setminus \mathcal I$, then we say that $B$ chooses between $\mathcal I$ and $\mathcal J$. We consider the following Ramsey-theoretic problem: Given several pairs $(\mathcal I_1,\mathcal J_1), (\mathcal I_2,\mathcal J_2), \dots, (\mathcal I_k,\mathcal J_k)$ of incompatible ideals on a set $X$, find some $A \subseteq X$ that chooses between as many of these pairs of ideals as possible. The main theorem is that for every $n \in \mathbb N$, there is some $I(n) \in \mathbb N$ such that given at least $I(n)$ pairs of incompatible ideals on any set $X$, there is some $A \subseteq X$ choosing between at least $n$ of them. This theorem is proved in two main steps. The first step is to identify a (purely finitary) problem in extremal combinatorics, and to show that our problem concerning ideals is equivalent to this combinatorial problem. The second step is to analyze the combinatorial problem in order to show that the number $I(n)$ described above exists, and to put bounds on it. We show $\textstyle \frac{1}{2}n \log_2 n - O(n) \,<\, I(n) \,<\, n \ln n + O(n).$ The upper bound is proved by considering a different but closely related combinatorial problem involving hypergraphs, which may be of independent interest. We also investigate some applications of this theorem to a problem concerning conditionally convergent series.

math.CO↗

arXiv2019

Linear operators with infinite entropy

We examine the chaotic behavior of certain continuous linear operators on infinite-dimensional Banach spaces, and provide several equivalent characterizations of when these operators have infinite topological entropy. For example, it is shown that infinite topological entropy is equivalent to non-zero topological entropy for translation operators on weighted Lebesgue function spaces. In particular, finite non-zero entropy is impossible for this class of operators, which answers a question raised by Yin and Wei.

math.DS↗

arXiv2019

The Rearrangement Number

How many permutations of the natural numbers are needed so that every conditionally convergent series of real numbers can be rearranged to no longer converge to the same sum? We define the \emph{rearrangement number}, a new cardinal characteristic of the continuum, as the answer to this question. We compare the rearrangement number with several natural variants, for example one obtained by requiring the rearranged series to still converge but to a new, finite limit. We also compare the rearrangement number with several well-studied cardinal characteristics of the continuum. We present some new forcing constructions designed to add permutations that rearrange series from the ground model in particular ways, thereby obtaining consistency results going beyond those that follow from comparisons with familiar cardinal characteristics. Finally, we deal briefly with some variants concerning rearrangements by a special sort of permutation and with rearranging some divergent series to become (conditionally) convergent.

math.LO↗

arXiv2019

The isomorphism class of the shift map

The \emph{shift map} $σ$ is the self-homeomorphism of $ω^* = βω\setminus ω$ induced by the successor function $n \mapsto n+1$ on $ω$. We prove that the isomorphism classes of $σ$ and $σ^{-1}$ cannot be separated by a Borel set in $\mathcal H(ω^*)$, the space of all self-homeomorphisms of $ω^*$ equipped with the compact-open topology. Van Douwen proved it is consistent for $σ$ and $σ^{-1}$ not to be isomorphic. Whether it is also consistent for them to be isomorphic is an open problem. The theorem stated above can be thought of as a counterpoint to van Douwen's result: while $σ$ and $σ^{-1}$ may not be isomorphic, there is no simple topological property that distinguishes them. As a relatively straightforward consequence of the main theorem, we deduce that $\mathsf{OCA}+\mathsf{MA}$ implies the set of continuous images of $σ$ fails to be Borel in $\mathcal H(ω^*)$. (Here a ``continuous image'' of $σ$ is meant in the sense of topological dynamics: any $h \in \mathcal H(ω^*)$ such that $q \circ σ= h \circ q$ for some continuous surjection $q: ω^* \to ω^*$.) This contrasts starkly with a recent theorem of the author showing that under $\mathsf{CH}$, the continuous images of $σ$ form a closed subset of $\mathcal H(ω^*)$.

math.GN↗

arXiv2019

Small cardinals and small Efimov spaces

We introduce and analyze a new cardinal characteristic of the continuum, the \emph{splitting number of the reals}, denoted $\mathfrak{s}(\mathbb R)$. This number is connected to Efimov's problem, which asks whether every infinite compact Hausdorff space must contain either a non-trivial convergent sequence, or else a copy of $β\mathbb N$.

math.LO↗

arXiv2018

Three conditionally convergent series

It is proved that given any three conditionally convergent series of real numbers, there is a single sequence of natural numbers such that each of the corresponding three subseries sums to either $\infty$ or $-\infty$. An example is provided to show that the analogous statement for four series is false.

math.CA↗

arXiv2018

Factoring a minimal ultrafilter into a thick part and a syndetic part

Let $S$ be an infinite discrete semigroup. The operation on $S$ extends uniquely to the Stone-Čech compactification $βS$ making $βS$ a compact right topological semigroup with $S$ contained in its topological center. As such, $βS$ has a smallest two sided ideal, $K(βS)$. An ultrafilter $p$ on $S$ is \emph{minimal} if and only if $p \in K(βS)$. We show that any minimal ultrafilter $p$ factors into a thick part and a syndetic part. That is, there exist filters $\mathcal F$ and $\mathcal G$ such that $\mathcal F$ consists only of thick sets, $\mathcal G$ consists only of syndetic sets, and $p$ is the unique ultrafilter containing $\mathcal F \cup \mathcal G$. Letting $L = \widehat{\mathcal F}$ and $C = \widehat{\mathcal G}$, the sets of ultrafilters containing $\mathcal F$ and $\mathcal G$ respectively, we have that $L$ is a minimal left ideal of $βS$, $C$ meets every minimal left ideal of $βS$ in exactly one point, and $L \cap C = \{p\}$. We show further that $K(βS)$ can be partitioned into relatively closed sets, each of which meets each minimal left ideal in exactly one point. With some weak cancellation assumptions on $S$, one has also that for each minimal ultrafilter $p$, $S^* \setminus \{p\}$ is not normal. In particular, if $p$ is a member of either of the disjoint sets $K(β\mathbb N , +)$ or $K(β\mathbb N , \cdot)$, then $\mathbb N^* \setminus \{p\}$ is not normal.

math.LO↗

arXiv2018

Universal flows and automorphisms of $\mathcal P(ω)/\mathrm{fin}$

We prove that for every countable discrete group $G$, there is a $G$-flow on $ω^*$ that has every $G$-flow of weight $\leq\! \aleph_1$ as a quotient. It follows that, under the Continuum Hypothesis, there is a universal $G$-flow of weight $\leq\!\mathfrak{c}$. Applying Stone duality, we deduce that, under \mathsf{CH}, there is a trivial automorphism $τ$ of $\mathcal P(ω)/\mathrm{fin}$ with every other automorphism embedded in it, which means that every other automorphism of $\mathcal P(ω)/\mathrm{fin}$ can be written as the restriction of $τ$ to a suitably chosen subalgebra.

math.GN↗

arXiv2018

The subseries number

Every conditionally convergent series of real numbers has a divergent subseries. How many subsets of the natural numbers are needed so that every conditionally convergent series diverges on the subseries corresponding to one of these sets? The answer to this question is defined to be the subseries number, a new cardinal characteristic of the continuum. This cardinal is bounded below by $\aleph_1$ and above by the cardinality of the continuum, but it is not provably equal to either. We define three natural variants of the subseries number, and compare them with each other, with their corresponding rearrangement numbers, and with several well-studied cardinal characteristics of the continuum. Many consistency results are obtained from these comparisons, and we obtain another by computing the value of the subseries number in the Laver model.

math.LO↗

arXiv2017

Which subsets of an infinite random graph look random?

Given a countable graph, we say a set $A$ of its vertices is \emph{universal} if it contains every countable graph as an induced subgraph, and $A$ is \emph{weakly universal} if it contains every finite graph as an induced subgraph. We show that, for almost every graph on $\mathbb N$, $(1)$ every set of positive upper density is universal, and $(2)$ every set with divergent reciprocal sums is weakly universal. We show that the second result is sharp (i.e., a random graph on $\mathbb N$ will almost surely contain non-universal sets with divergent reciprocal sums) and, more generally, that neither of these two results holds for a large class of partition regular families.

math.CO↗

arXiv2016

Shift-preserving maps on $ω^*$

The shift map $σ$ on $ω^*$ is the continuous self-map of $ω^*$ induced by the function $n \mapsto n+1$ on $ω$. Given a compact Hausdorff space $X$ and a continuous function $f: X \rightarrow X$, we say that $(X,f)$ is a quotient of $(ω^*,σ)$ whenever there is a continuous surjection $Q: ω^* \to X$ such that $Q \circ σ= f \circ Q$. Our main theorem states that if the weight of $X$ is at most $\aleph_1$, then $(X,f)$ is a quotient of $(ω^*,σ)$ if and only if $f$ is weakly incompressible (which means that no nontrivial open $U \subseteq X$ has $f(\bar{U}) \subseteq U$). Under CH, this gives a complete characterization of the quotients of $(ω^*,σ)$ and implies, for example, that $(ω^*,σ^{-1})$ is a quotient of $(ω^*,σ)$. In the language of topological dynamics, our theorem states that a dynamical system of weight $\aleph_1$ is an abstract $ω$-limit set if and only if it is weakly incompressible. We complement these results by proving $(1)$ our main theorem remains true when $\aleph_1$ is replaced by any $κ< \mathfrak{p}$, $(2)$ consistently, the theorem becomes false if we replace $\aleph_1$ by $\aleph_2$, and $(3)$ OCA+MA implies that $(ω^*,σ^{-1})$ is not a quotient of $(ω^*,σ)$.

math.GN↗

arXiv2015

$G_δ$ semifilters and $ω^*$

The ultrafilters on the partial order $([ω]^ω,\subseteq^*)$ are the free ultrafilters on $ω$, which constitute the space $ω^*$, the Stone-Cech remainder of $ω$. If $U$ is an upperset of this partial order (i.e., a semifilter), then the ultrafilters on $U$ correspond to closed subsets of $ω^*$ via Stone duality. If, in addition, $U$ is sufficiently "simple" (more precisely, $G_δ$ as a subset of $2^ω$), we show that $U$ is similar to $[ω]^ω$ in several ways. First, $\mathfrak{p}_U = \mathfrak{t}_U = \mathfrak{p}$ (this extends a result of Malliaris and Shelah). Second, if $\mathfrak{d} = \mathfrak{c}$ then there are ultrafilters on $U$ that are also $P$-filters (this extends a result of Ketonen). Third, there are ultrafilters on $U$ that are weak $P$-filters (this extends a result of Kunen). By choosing appropriate $U$, these similarity theorems find applications in dynamics, algebra, and combinatorics. Most notably, we will prove that $(ω^*,+)$ contains minimal left ideals that are also weak $P$-sets.

math.LO↗

arXiv2015

Ideals and idempotents in the uniform ultrafilters

If $S$ is a discrete semigroup, then $βS$ has a natural, left-topological semigroup structure extending $S$. Under some very mild conditions, $U(S)$, the set of uniform ultrafilters on $S$, is a two-sided ideal of $βS$, and therefore contains all of its minimal left ideals and minimal idempotents. We find some very general conditions under which $U(S)$ contains prime minimal left ideals and left-maximal idempotents. If $S$ is countable, then $U(S) = S^*$, and a special case of our main theorem is that if a countable discrete semigroup $S$ is a weakly cancellative and left-cancellative, then $S^*$ contains prime minimal left ideals and left-maximal idempotents. We will provide examples of weakly cancellative semigroups where these conclusions fail, thus showing that this result is sharp.

math.RA↗

arXiv2015

From Haar to Lebesgue via Domain Theory, Revised version

If ${\mathcal C}\simeq 2^{\mathbb N}$ denotes the Cantor set realized as the infinite product of two-point groups, then a folklore result says the Cantor map from ${\mathcal C}$ into $[0,1]$ sends Haar measure to Lebesgue measure on the interval. In fact, ${\mathcal C}$ admits many distinct topological group structures. In this note, we show that the Haar measures induced by these distinct group structures are share this property. We prove this by showing that Haar measure for any group structure is the same as Haar measure induced by a related abelian group structure. Moreover, each abelian group structure on ${\mathcal C}$ supports a natural total order that determines a map onto the unit interval that is monotone, and hence sends intervals in ${\mathcal C}$ to subintervals of the unit interval. Using techniques from domain theory, we show this implies this map sends Haar measure on ${\mathcal C}$ to Lebesgue measure on the interval, and we then use this to contract a Borel isomorphism between any two group structures on ${\mathcal C}$.

math.FA↗