Slawomir Solecki

University of Illinois, Urbana-Champaign | UIUC · Department of Mathematics

We show that, for a generic measure preserving transformation T, the closed group generated by T is not isomorphic to the topological group \(L^0(\lambda , {{\mathbb {T}}})\) of all Lebesgue measurable functions from [0, 1] to \({\mathbb {T}}\) (taken with pointwise multiplication and the topology of convergence in measure). This result answers a q...

We compute the spectral form of the Koopman representation induced by a natural boolean action of $L^0(\lambda, {\mathbb T})$ identified earlier by the authors. Our computation establishes the sharpness of the constraints on spectral forms of Koopman representations of $L^0(\lambda, {\mathbb T})$ previously found by the second author.

We show that, for a generic measure preserving transformation $T$, the closed group generated by $T$ is not isomorphic to the topological group $L^0(\lambda, {\mathbb T})$ of all Lebesgue measurable functions from $[0,1]$ to $\mathbb T$ (taken with multiplication and the topology of converegence in measure). This result answers a question of Glasne...

Exhibiting a new type of measure concentration, we prove uniform concentration bounds for measurable Lipschitz functions on product spaces, where Lipschitz is taken with respect to the metric induced by a weighted covering of the index set of the product. Our proof combines the Herbst argument with an analogue of Shearer's lemma for differential en...

Exhibiting a new type of measure concentration, we prove uniform concentration bounds for measurable Lipschitz functions on product spaces, where Lipschitz is taken with respect to the metric induced by a weighted covering of the index set of the product. Our proof combines the Herbst argument with an entropic version of the weighted Loomis--Whitne...

We study ideals $\mathcal{I}$ on $\mathbb N$ satisfying the following Baire-type property: if $X$ is a complete metric space and $\{X_{A}: A \in \mathcal{I} \}$ is a family of nowhere dense subsets of $X$ with $X_{A} \subset X_{B}$ whenever $A \subset B$, then $\bigcup_{A \in \mathcal{I}}X_{A} \neq X$. We give several characterizations and determin...

We prove a Ramsey theorem for finite sets equipped with a partial order and a fixed number of linear orders extending the partial order. This is a common generalization of two recent Ramsey theorems due to Soki\'c. As a bonus, our proof gives new arguments for these two results.

We prove that for a generic measure preserving transformation $T$, the closed group generated by $T$ is a continuous homomorphic image of a closed linear subspace of $L_0(\lambda,{\mathbb R})$, where $\lambda$ is Lebesgue measure, and that the closed group generated by $T$ contains an increasing sequence of finite dimensional toruses whose union is...

I will give a presentation of an abstract approach to finite Ramsey theory found in an earlier paper of mine. I will prove from it a common generalization of Deuber's Ramsey theorem for regular trees and a recent Ramsey theorem of Jasinski for boron tree structures. This generalization appears to be new. I will also show, in exercises, how to deduc...

The space of Lascar strong types, on some sort and relative to a given first order theory T, is in general not a compact Hausdorff space. This paper has at least three aims. First to show that spaces of Lascar strong types and other related spaces such as the Lascar group, have well-defined Borel cardinalities. The second is to compute the Borel ca...

We prove the direct structural Ramsey theorem for structures with relations as well as functions. The result extends the theorem of Abramson and Harrington and of Nešetřil and Rödl.

We will demonstrate that if M is an uncountable compact metric space, then there is an action of the Polish group of all continuous functions from M to U(1) on a separable probability algebra which preserves the measure and yet does not admit a point realization in the sense of Mackey. This is achieved by exhibiting a strong form of ergodicity of t...

We present a short and elementary proof of isometric uniqueness of the Gurarii space.

We give an abstract approach to finite Ramsey theory and prove a general Ramsey-type theorem. We deduce from it a self-dual Ramsey theorem, which is a new result naturally generalizing both the classical Ramsey theorem and the dual Ramsey theorem of Graham and Rothschild. In fact, we recover the pure finite Ramsey theory from our general Ramsey-typ...

We investigate the structure of G δ ideals of compact sets. We define a class of G δ ideals of compact sets that, on the one hand, avoids certain phenomena present among general G δ ideals of compact sets and, on the other hand, includes all naturally occurring G δ ideals of compact sets. We prove structural theorems for ideals in this class, and w...

We prove a generalization of Promel's theorem to flnite structures with both relations and functions. The classical Ramsey theorem (11) has been generalized in two (among other) important, independent directions. First, Graham and Rothschild (3) extended Ramsey's theorem from flnite sets to flnite parameter sets, objects more general than sets, who...

We find a characterization of those Polish ultrametric spaces on which each Baire one function is first return recoverable. The notion of pseudo-convergence originating in the theory of valuation fields plays a crucial role in the characterization.

The paper studies the structure of the homogeneous space G/H, for G a Polish group and H < G a Borel, not necessarily closed subgroup of G, from the point of view of the theory of definable equiv-alence relations. It makes a connection between the complexity of the natural coset equivalence relation associated with G/H and Polishability of H, that...

We prove that there is a G- æ-ideal of compact sets which is strictly above NWD in the Tukey order. Here NWD is the collection of all compact nowhere dense subsets of the Cantor set. This answers a question of Louveau and Velickovic asked in (4).

We show that every locally compact Polish group is iso-morphic to the isometry group of a proper separable metric space. This answers a question of Gao and Kechris. We also analyze the natural action of the isometry group of a separable ultrametric space on the space. This leads us to a structure theorem representing an arbitrary separable ultramet...

The paper has two objectives. On the one hand, we study left Haar null sets—a measure theoretic notion of smallness on Polish, not necessarily locally compact, groups. On the other hand, we introduce and investigate two classes of Polish groups which are closely related to this notion and to amenability. We show that left Haar null sets form a σ-id...

The aim of the present work is to develop a dualization of the Frassé limit construction from Model Theory and to indicate its surprising connections with the pseudo-arc. As corollaries of general results on the dual Frassé lim-its, we obtain Mioduszewski's theorem on surjective universality of the pseudo-arc among chainable continua and a theorem...

We prove several trichotomy results for ideals of compact sets. Typically, we show that a “sufficiently rich” universally Baire ideal is either Π3 0-hard. or Σ3 0-hard. or else a σ-ideal.

We prove that arbitrary homomorphisms from one of the groups ${\rm Homeo}(\ca)$, ${\rm Homeo}(\ca)^\N$, ${\rm Aut}(\Q,<)$, ${\rm Homeo}(\R)$, or ${\rm Homeo}(S^1)$ into a separable group are automatically continuous. This has consequences for the representations of these groups as discrete groups. For example, it follows, in combination with a resu...

We find two F σ δ F_{\sigma \delta } ideals on N \mathbb N neither of which is F σ F_\sigma whose quotient Boolean algebras are homogeneous but nonisomorphic. This solves a problem of Just and Krawczyk (1984).

We show that each non-compact Polish group admits a continuous action on a Polish space with non-smooth orbit equivalence relation. We actually construct a free such action. Thus for a Polish group compactness is equivalent to all continuous free actions of this group being smooth. This answers a question of Kechris. We also establish results relat...

This paper presents a logical system in which various group-level epistemic actions are incorporated into the object language. That is, we consider the standard modeling of knowledge among a set of agents by multimodal Kripke structures. One might want to consider actions that take place, such as announcements to groups privately, announcements wit...

This paper presents a logical system in which various group-level epistemic actions are incorporated into the object language. That is, we consider the standard modeling of knowledge among a set of agents by multi-modal Kripke structures. One might want to consider actions that take place, such as announcements to groups privately, announcements wi...

We study the structure of analytic ideals of subsets of the natural numbers. For example, we prove that for an analytic ideal I, either the ideal {X ⊂ (Ω × Ω: ⊂En X ⊂({0, 1,…,n} × Ω } is Rudin-Keisler below I, or I is very simply induced by a lower semicontinuous submeasure. Also, we show that the class of ideals induced in this manner by lsc subme...

We extend the original Glimm-Eros theorem for locally compact groups to a class of Polish groups including the nilpotent ones and those with an invariant metric. For this class we thereby obtain the topological Vaught conjecture. 0. Preface In this paper we consider equivalence relations induced by Polish groups acting continuously on Polish spaces...

We prove that every analytic set in $^\omega\omega \times ^\omega\omega$ with $\sigma$-bounded sections has a not $\sigma$-bounded closed free set. We show that this result is sharp. There exists a closed set with bounded sections which has no dominating analytic free set, and there exists a closed set with non-dominating sections which does not ha...

We study in this paper graph coloring problems in the context of descriptive set theory. We consider graphs G=(X, R), where the vertex set X is a standard Borel space (i.e., a complete separable metrizable space equipped with its σ-algebra of Borel sets), and the edge relation R ⊆ X^2 is "definable", i.e., Borel, analytic, co-analytic, etc. A Bore...

We establish dichotomy results concerning the structure of Baire class 1 functions. We consider decompositions of Baire class 1 functions into continuous functions and into continuous functions with closed domains. Dichotomy results for both of them are proved: a Baire class 1 function decomposes into countably many countinuous functions, or else c...

We investigate the possibility of making all functions in Baire class $\alpha$, $\alpha

We prove that if (G, ·) is a group with a metric, separable, and Baire topology such that h → g · h is continuous for all gϵG and g → g · h is Baire measurable for all hϵG, then (G, ) is a topological group. Several consequences of this result concerning free actions of standard Borel groups are established.

§1. Introduction . Ideals and filters of subsets of natural numbers have been studied by set theorists and topologists for a long time. There is a vast literature concerning various kinds of ultrafilters (or, dually, maximal ideals). There is also a substantial interest in nicely definable (Borel, analytic) ideals—these by old results of Sierpiński...

We give an algebraic characterization of those sequences (H n ) of countable abelian groups for which the equivalence relations induced by Borel (or, equivalently, continuous) actions of H 0 ×H 1 ×H 2 ×⋯ are Borel. In particular, the equivalence relations induced by Borel actions of H ω , H countable abelian, are Borel iff H≃⨁ p (F p ×ℤ(p ∞ ) n p )...

LetI be a σ-ideal on a Polish space such that each set fromI is contained in a Borel set fromI. We say thatI fails to fulfil theΣ 1 1 countable chain condition if there is aΣ 1 1 equivalence relation with uncountably many equivalence classes none of which is inI. Assuming definable determinacy, we show that if the family of Borel sets fromI i...

We prove that for every family $I$ of closed subsets of a Polish space each $\Sigma^1_1$ set can be covered by countably many members of $I$ or else contains a nonempty $\Pi^0_2$ set which cannot be covered by countably many members of $I$. We prove an analogous result for $\kappa$-Souslin sets and show that if $A^\sharp$ exists for any $A \subset...

For A⊂2 ω and X⊂ω consider an infinite game Γ(A,X) in which two players I and II choose c n ∈{0,1}. c n is chosen by I if n∈X and by II if n∈ω∖X. I wins if (c 0 ,c 1 ,c 2 ,⋯)∈A. We analyze connections between A and the family of all sets X⊂ω for which I has a winning strategy in Γ(A,X). Certain similarities and differences appear if one formulates...

Assume a group G acts on a set. Given a subgroup H of G, by an H-selector we mean a selector of the set of all orbits of H. We investigate measurability properties of H-selectors with respect lo G-invariant measures.

We discuss in the paper the following problem: Given a function in a given Baire class, into "how many" (in terms of cardinal numbers) functions of lower classes can it be decomposed? The decomposition is understood here in the sense of the set-theoretical union.

A theorem is proved that the subgroup of n generated by a semicontinuum containing the origin is a linear subspace of n. It is the answer to the question of J.E. Keesling and D.C. Wilson.

We show that L0(,H ) is extremely amenable for any dif- fused submeasure and any solvable compact group H. This extends re- sults of Herer-Christensen and of Glasner and Furstenberg-Weiss. Proofs of these earlier results used spectral theory or concentration of measure. Our argument is based on a new Ramsey theorem proved using ideas com- ing from...

We nd two F ideals onN neither of which is F whose quotient Boolean algebras are homogeneous but nonisomorphic. This solves a problem of Just and Krawczyk (1984).

We give a short proof of the theorem that a closed subgroup of a countable product of second countable Lie groups is pro-Lie.