Homogeneous Borel Sets of Ambiguous Class Two

We describe and characterize all homogeneous subsets of the Cantor set which are both an ${F_{\sigma \delta }}$ and a ${G_{\delta \sigma }}$; it turns out that there are ${\omega _1}$ such spaces.

e-4 - Special Metrics

Let C 1 C_{1} , C 2 C_{2} be Cantor sets embedded in the real line, and let τ 1 \tau _{1} , τ 2 \tau _{2} be their respective thicknesses. If τ 1 τ 2 > 1 \tau _{1}\tau _{2}>1 , then it is well known that the difference set C 1 − C 2 C_{1}-C_{2} is a disjoint union of closed intervals. B. Williams showed that for some t ∈ int ⁡ ( C 1 − C 2 ) t\in \operatorname {int} (C_{1}-C_{2}) , it may be that C 1 ∩ ( C 2 + t ) C_{1}\cap (C_{2}+t) is as small as a single point. However, the author previously showed that generically, the other extreme is true; C 1 ∩ ( C 2 + t ) C_{1}\cap (C_{2}+t) contains a Cantor set for all t t in a generic subset of C 1 − C 2 C_{1}-C_{2} . This paper shows that small intersections of thick Cantor sets are also rare in the sense of Lebesgue measure; if τ 1 τ 2 > 1 \tau _{1}\tau _{2}>1 , then C 1 ∩ ( C 2 + t ) C_{1}\cap (C_{2}+t) contains a Cantor set for almost all t t in C 1 − C 2 C_{1}-C_{2} .

Intersection of triadic Cantor sets with their translates— I. Fundamental properties

A subset X of a Polish group G is called Haar null if there exist a Borel set B ⊃ X and Borel probability measure μ on G such that μ(gBh) = 0 for every g; h ∊ G. We prove that there exist a set X ⊂ R that is not Lebesgue null and a Borel probability measure μ such that μ (X + t) = 0 for every t ∊ R. This answers a question from David Fremlin’s problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set B. (The answer was already known assuming the Continuum Hypothesis.)This result motivates the following Baire category analogue. It is consistent with ZFC that there exist an abelian Polish group G and a Cantor set C ⊂ G such that for every non-meagre set X ⊂ G there exists a t ∊ G such that C ∩ (X + t) is relatively non-meagre in C. This essentially generalizes results of Bartoszyński and Burke–Miller.

Various parametrization theorems are proved. In particular the following is shown: Let B B be a Borel subset of I × I I \times I (where I = [ 0 , 1 ] I = [0,\,1] ) with uncountable vertical sections. Let ∑ ∪ ˙ N \sum \dot \cup N be the discrete (topological) union of ∑ \sum , the space of irrationals, and N N , the set of natural numbers with discrete topology. Then there is a map f : I × ( ∑ ∪ ˙ N ) → I f:I \times (\sum \dot \cup N) \to I measurable with respect to the product of the analytic σ \sigma -field on I I (that is, the smallest σ \sigma -field on I I containing the analytic sets) and the Borel σ \sigma -field on ∑ ∪ ˙ N \sum \dot \cup N such that f ( t , ⋅ ) : ∑ ∪ ˙ N → I f(t,\,\, \cdot ):\,\sum \dot \cup N \to I is a one-one continuous map of ∑ ∪ ˙ N \sum \dot \cup N onto { x : ( t , x ) ∈ B } \{ x:(t,\,x) \in B\} for each t ∈ T t \in T . This answers a question of Cenzer and Mauldin.

LetE be a homogeneous compact set, for instance a Cantor set of positive length. Further, let σ be a positive measure with supp(σ)=E. Under the condition that the absolutely continuous part of σ satisfies a Szego-type condition, we give an asymptotic representation, on and off the support, for the polynomials orthonomal with respect to σ. For the special case thatE consists of a finite number of intervals and that σ has no singular component, this is a well-known result of Widom. IfE=[a,b], it becomes a classical result due to Szego; and in case that there appears in addition a singular component, it is due to Kolmogorov-krein. In fact, the results are presented for the more general case that the orthogonality measure may have a denumerable set of mass-points outside ofE which are supposed to accumulate only onE and to satisfy (together with the zeros of the associated Stieltjes function) the free-interpolation Carleson-type condition. Up to the case of a finite number of mass points, this is even new for the single interval case. Furthermore, as a byproduct of our representations, we obtain that the recurrence coefficients of the orthonormal polynomials behave asymptotically almost periodic. In other words, the Jacobi matrices associated with the above discussed orthonomal polynomials are compact perturbations of a onesided restriction of almost periodic Jacobi matrices with homogeneous spectrum. Our main tool is a theory of Hardy spaces of character-automorphic functions and forms on Riemann surfaces of Widom type; we use also some ideas of scattering theory for one-dimensional Schrodinger equations.

Topological characterizations of all zero-dimensional homogeneous absolute Borel sets are obtained; it turns out that there are ω 1 {\omega _1} such spaces. We use results from game theory—particularly, about Wadge classes.

A metric space X has the unique midset property if there is a topology-preserving metric d on X such that for every pair of distinct points x, y there is one and only one point p such that d(x, p) = d(y, p). The following are proved. (1) The discrete space with cardinality n has the unique midset property if and only if n ≠ 2, 4 and n ⩽ c, where c is the cardinality of the continuum. (2) If D is a discrete space with cardinality not greater than c, then the countable power DN of D has the unique midset property. In particular, the Cantor set and the space of irrational numbers have the unique midset property. A finite discrete space with n points has the unique midset property if and only if there is an edge colouring φ of the complete graph Kn such that for every pair of distinct vertices x, y there is one and only one vertex p such that φ(xp) = φ(yp). Let ump(Kn) be the smallest number of colours necessary for such a colouring of Kn. The following are proved. (3) For each k ⩾ 0, ump(K2k+1) = k. (4) For each k ⩾ 3, k ⩽ ump(K2k) ⩽ 2k−1.

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Asymptotic order of quantization for Cantor distributions in terms of Euler characteristic, Hausdorff and Packing measure

§1. I will start with a quick definition of descriptive set theory: It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Such spaces are usually calledPolish spaces. Typical examples are ℝn, ℂn, (separable) Hilbert space and more generally all separable Banach spaces, theCantor space2ℕ, theBaire spaceℕℕ, the infinite symmetric groupS∞, the unitary group (of the Hilbert space), the group of measure preserving transformations of the unit interval, etc.In this theory sets are classified in hierarchies according to the complexity of their definitions and the structure of sets in each level of these hierarchies is systematically analyzed. In the beginning we have theBorelsets in Polish spaces, obtained by starting with the open sets and closing under the operations of complementation and countable unions, and the correspondingBorel hierarchy(sets). After this come theprojective sets, obtained by starting with the Borel sets and closing under the operations of complementation and projection, and the correspondingprojective hierarchy(sets).There are also transfinite extensions of the projective hierarchy and even much more complex definable sets studied in descriptive set theory, but I will restrict myself here to Borel and projective sets, in fact just those at the first level of the projective hierarchy, i.e., theBorel(),analytic() andcoanalytic() sets.

We construct an amenable action Φ of a non-amenable group Γ on a discrete space. This action extends to a minimal topological action Φ of Γ on a Cantor set C. We show that Φ is non-uniquely ergodic and furthermore there exist ergodic invariant measures μ1 and μ2 such that (Φ,C,μ1) and (Φ,C,μ2) are not orbit equivalent measurable equivalence relations. This also provides an instance of the failure of equivalence between the notions of “global” and “local” amenability for countable equivalence relations.

Stable intersection of affine Cantor sets defined by two expanding maps

ABSTRACTLet X be a zero‐dimensional first category absolute Borel set of ambiguous class two. Then X can be given the structure of a topological group if and only if X is homogeneous and X=X x X. Thus, within the ambiguous class two, zero‐dimensional groups only occur at the low levels, and the indecomposable levels of the hierarchy of difference classes DαΣ02 in 2ω.