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Pandelis Dodos

Publications and source records attributed to Pandelis Dodos.

At least 37 records · Page 2Linked to original sources
arXiv2013

A density version of the Halpern-Läuchli theorem

We prove a density version of the Halpern-Läuchli Theorem. This settles in the affirmative a conjecture of R. Laver. Specifically, let us say that a tree $T$ is homogeneous if $T$ has a unique root and there exists an integer $b\meg 2$ such that every $t\in T$ has exactly $b$ immediate successors. We show that for every $d\meg 1$ and every tuple $(T_1,...,T_d)$ of homogeneous trees, if $D$ is a subset of the level product of $(T_1,...,T_d)$ satisfying \[ \limsup_{n\to\infty} \frac{|D\cap \big(T_1(n)\times ... \times T_d(n)\big)|}{|T_1(n)\times ... \times T_d(n)|}>0\] then there exist strong subtrees $(S_1, ..., S_d)$ of $(T_1,...,T_d)$ having common level set such that the level product of $(S_1,...,S_d)$ is a subset of $D$.

math.CO↗
arXiv2013

Measurable events indexed by products of trees

A tree $T$ is said to be homogeneous if it is uniquely rooted and there exists an integer $b\meg 2$, called the branching number of $T$, such that every $t\in T$ has exactly $b$ immediate successors. A vector homogeneous tree $\mathbf{T}$ is a finite sequence $(T_1,...,T_d)$ of homogeneous trees and its level product $\otimes\mathbf{T}$ is the subset of the cartesian product $T_1\times ...\times T_d$ consisting of all finite sequences $(t_1,...,t_d)$ of nodes having common length. We study the behavior of measurable events in probability spaces indexed by the level product $\otimes\mathbf{T}$ of a vector homogeneous tree $\mathbf{T}$. We show that, by refining the index set to the level product $\otimes\mathbf{S}$ of a vector strong subtree $\bfcs$ of $\mathbf{S}$, such families of events become highly correlated. An analogue of Lebesgue's density Theorem is also established which can be considered as the "probabilistic" version of the density Halpern--Läuchli Theorem.

math.CO↗
arXiv2013

A simple proof of the density Hales-Jewett theorem

We give a purely combinatorial proof of the density Hales--Jewett Theorem that is modeled after Polymath's proof but is significantly simpler. In particular, we avoid the use of the equal-slices measure and work exclusively with the uniform measure.

math.CO↗
arXiv2012

Measurable events indexed by trees

A tree $T$ is said to be homogeneous if it is uniquely rooted and there exists an integer $b\geq 2$, called the branching number of $T$, such that every $t\in T$ has exactly $b$ immediate successors. We study the behavior of measurable events in probability spaces indexed by homogeneous trees. Precisely, we show that for every integer $b\geq 2$ and every integer $n\geq 1$ there exists an integer $q(b,n)$ with the following property. If $T$ is a homogeneous tree with branching number $b$ and $\{A_t:t\in T\}$ is a family of measurable events in a probability space $(Ω,Σ,μ)$ satisfying $μ(A_t)\geqε>0$ for every $t\in T$, then for every $0<θ<ε$ there exists a strong subtree $S$ of $T$ of infinite height such that for every non-empty finite subset $F$ of $S$ of cardinality $n$ we have \[ μ\Big(\bigcap_{t\in F} A_t\Big) \meg θ^{q(b,n)}. \] In fact, we can take $q(b,n)= \big((2^b-1)^{2n-1}-1\big)\cdot(2^b-2)^{-1}$. A finite version of this result is also obtained.

math.CO↗
arXiv2012

Dense subsets of products of finite trees

We prove a "uniform" version of the finite density Halpern-Läuchli Theorem. Specifically, we say that a tree $T$ is homogeneous if it is uniquely rooted and there is an integer $b\geq 2$, called the branching number of $T$, such that every $t\in T$ has exactly $b$ immediate successors. We show the following. For every integer $d\geq 1$, every $b_1,...,b_d\in\mathbb{N}$ with $b_i\geq 2$ for all $i\in\{1,...,d\}$, every integer $k\meg 1$ and every real $0<ε\leq 1$ there exists an integer $N$ with the following property. If $(T_1,...,T_d)$ are homogeneous trees such that the branching number of $T_i$ is $b_i$ for all $i\in\{1,...,d\}$, $L$ is a finite subset of $\mathbb{N}$ of cardinality at least $N$ and $D$ is a subset of the level product of $(T_1,...,T_d)$ satisfying \[|D\cap \big(T_1(n)\times ...\times T_d(n)\big)| \geq ε|T_1(n)\times ...\times T_d(n)|\] for every $n\in L$, then there exist strong subtrees $(S_1,...,S_d)$ of $(T_1,...,T_d)$ of height $k$ and with common level set such that the level product of $(S_1,...,S_d)$ is contained in $D$. The least integer $N$ with this property will be denoted by $UDHL(b_1,...,b_d|k,ε)$. The main point is that the result is independent of the position of the finite set $L$. The proof is based on a density increment strategy and gives explicit upper bounds for the numbers $UDHL(b_1,...,b_d|k,ε)$.

math.CO↗
arXiv2011

Operators whose dual has non-separable range

Let $X$ and $Y$ be separable Banach spaces and $T:X\to Y$ be a bounded linear operator. We characterize the non-separability of $T^*(Y^*)$ by means of fixing properties of the operator $T$.

math.FA↗
arXiv2010

Dichotomies of the set of test measures of a Haar-null set

We prove that if $X$ is a Polish space and $F$ is a face of $P(X)$ with the Baire property, then $F$ is either a meager or a co-meager subset of $P(X)$. As a consequence we show that for every abelian Polish group $X$ and every analytic Haar-null set $A\subseteq X$, the set of test measures $T(A)$ of $A$ is either meager or co-meager. We characterize the non-locally-compact groups as the ones for which there exists a closed Haar-null set $F\subseteq X$ with $T(F)$ is meager. Moreover, we answer negatively a question of J. Mycielski by showing that for every non-locally-compact abelian Polish group and every $σ$-compact subgroup $G$ of $X$ there exists a $G$-invariant $F_σ$ subset of $X$ which is neither prevalent nor Haar-null.

math.FA↗
arXiv2010

The Steinhaus property and Haar-null sets

It is shown that if $G$ is an uncountable Polish group and $A\subseteq G$ is a universally measurable set such that $A^{-1}A$ is meager, then the set $T_l(A)=\{μ\in P(G): μ(gA)=0 \text{for all} g\in G\}$ is co-meager. In particular, if $A$ is analytic and not left Haar-null, then $1\in\mathrm{Int}(A^{-1}AA^{-1}A)$.

math.FA↗
arXiv2010

On strictly singular operators between separable Banach spaces

Let $X$ and $Y$ be separable Banach spaces and denote by $\sss\sss(X,Y)$ the subset of $\llll(X,Y)$ consisting of all strictly singular operators. We study various ordinal ranks on the set $\sss\sss(X,Y)$. Our main results are summarized as follows. Firstly, we define a new rank $\rs$ on $\sss\sss(X,Y)$. We show that $\rs$ is a co-analytic rank and that dominates the rank $\varrho$ introduced by Androulakis, Dodos, Sirotkin and Troitsky [Israel J. Math., 169 (2009), 221-250]. Secondly, for every $1\leq p<+\infty$ we construct a Banach space $Y_p$ with an unconditional basis such that $\sss\sss(\ell_p, Y_p)$ is a co-analytic non-Borel subset of $\llll(\ell_p,Y_p)$ yet every strictly singular operator $T:\ell_p\to Y_p$ satisfies $\varrho(T)\leq 2$. This answers a question of Argyros.

math.FA↗
arXiv2008

On classes of Banach spaces admitting "small" universal spaces

We characterize those classes $\ccc$ of separable Banach spaces admitting a separable universal space $Y$ (that is, a space $Y$ containing, up to isomorphism, all members of $\ccc$) which is not universal for all separable Banach spaces. The characterization is a byproduct of the fact, proved in the paper, that the class $\mathrm{NU}$ of non-universal separable Banach spaces is strongly bounded. This settles in the affirmative the main conjecture form \cite{AD}. Our approach is based, among others, on a construction of $\llll_\infty$-spaces, due to J. Bourgain and G. Pisier. As a consequence we show that there exists a family $\{Y_ξ:ξ<ω_1\}$ of separable, non-universal, $\llll_\infty$-spaces which uniformly exhausts all separable Banach spaces. A number of other natural classes of separable Banach spaces are shown to be strongly bounded as well.

math.FA↗
arXiv2008

Unconditional basic sequences in spaces of large density

We study the problem of the existence of unconditional basic sequences in Banach spaces of high density. We show, in particular, the relative consistency with GCH of the statement that every Banach space of density $\aleph_ω$ contains an unconditional basic sequence.

math.FA↗
arXiv2008

Definability under duality

It is shown that if $A$ is an analytic class of separable Banach spaces with separable dual, then the set $A^*=\{Y:\exists X\in A \text{with} Y\cong X^*\}$ is analytic. The corresponding result for pre-duals is false.

math.FA↗
arXiv2008

Codings of separable compact subsets of the first Baire class

Let $X$ be a Polish space and $K$ a separable compact subset of the first Baire class on $X$. For every sequence $\bs$ dense in $\kk$, the descriptive set-theoretic properties of the set \[ \lbf=\{L\in[\nn]: (f_n)_{n\in L} \text{is pointwise convergent}\} \] are analyzed. It is shown that if $K$ is not first countable, then $\lbf$ is $\PB^1_1$-complete. This can also happen even if $K$ is a pre-metric compactum of degree at most two, in the sense of S. Todorcevic. However, if $K$ is of degree exactly two, then $\lbf$ is always Borel. A deep result of G. Debs implies that $\lbf$ contains a Borel cofinal set and this gives a tree-representation of $\kk$. We show that classical ordinal assignments of Baire-1 functions are actually $\PB^1_1$-ranks on $\kk$. We also provide an example of a $\SB^1_1$ Ramsey-null subset $A$ of $[\nn]$ for which there does not exist a Borel set $B\supseteq A$ such that the difference $B\setminus A$ is Ramsey-null.

math.LO↗
arXiv2008

On pairs of definable orthogonal families

We introduce the notion of an M-family of infinite subsets of $\nn$ which is implicitly contained in the work of A. R. D. Mathias. We study the structure of a pair of orthogonal hereditary families $\aaa$ and $\bbb$, where $\aaa$ is analytic and $\bbb$ is $C$-measurable and an M-family.

math.LO↗