A generalization of Zeeman’s family

Michał Sierakowski

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Displaying similar documents to “A generalization of Zeeman’s family”

Difference functions of periodic measurable functions

Tamás Keleti (1998)

Fundamenta Mathematicae

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We investigate some problems of the following type: For which sets H is it true that if f is in a given class ℱ of periodic functions and the difference functions $Δ_{h} f (x) = f (x + h) - f (x)$ are in a given smaller class G for every h ∈ H then f itself must be in G? Denoting the class of counter-example sets by ℌ(ℱ,G), that is, $ℌ (ℱ, G) = H \subset ℝ / ℤ : (\exists f \in ℱ G) (\forall h \in H) Δ_{h} f \in G$ , we try to characterize ℌ(ℱ,G) for some interesting classes of functions ℱ ⊃ G. We study classes of measurable functions on the circle group $𝕋 = ℝ / ℤ$ that are invariant for changes on null-sets...

How to recognize a true Σ^0_3 set

Etienne Matheron (1998)

Fundamenta Mathematicae

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Let X be a Polish space, and let ${(A_{p})}_{p \in ω}$ be a sequence of $G_{δ}$ hereditary subsets of K(X) (the space of compact subsets of X). We give a general criterion which allows one to decide whether $\cup_{p \in ω} A_{p}$ is a true $\sum_{3}^{0}$ subset of K(X). We apply this criterion to show that several natural families of thin sets from harmonic analysis are true $\sum_{3}^{0}$ .

On sums of two cubes: an Ω₊-estimate for the error term

M. Kühleitner, W. G. Nowak, J. Schoissengeier, T. D. Wooley (1998)

Acta Arithmetica

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The arithmetic function $r_{k} (n)$ counts the number of ways to write a natural number n as a sum of two kth powers (k ≥ 2 fixed). The investigation of the asymptotic behaviour of the Dirichlet summatory function of $r_{k} (n)$ leads in a natural way to a certain error term $P_{_{k}} (t)$ which is known to be $O (t^{1 / 4})$ in mean-square. In this article it is proved that $P_{₃} (t) = Ω ₊ (t^{1 / 4} {(l o g l o g t)}^{1 / 4})$ as t → ∞. Furthermore, it is shown that a similar result would be true for every fixed k > 3 provided that a certain set of algebraic numbers contains a...

Strongly almost disjoint familes, revisited

A. Hajnal, Istvan Juhász, Saharon Shelah (2000)

Fundamenta Mathematicae

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The relations M(κ,λ,μ) → B [resp. B(σ)] meaning that if $A \subset {[κ]}^{λ}$ with |A|=κ is μ-almost disjoint then A has property B [resp. has a σ-transversal] had been introduced and studied under GCH in [EH]. Our two main results here say the following: Assume GCH and let ϱ be any regular cardinal with a supercompact [resp. 2-huge] cardinal above ϱ. Then there is a ϱ-closed forcing P such that, in $V^{P}$ , we have both GCH and $M (ϱ^{(+ ϱ + 1)}, ϱ^{+}, ϱ) ↛ B$ [resp. $M (ϱ^{(+ ϱ + 1)}, λ, ϱ) ↛ B (ϱ^{+})$ for all $λ \leq ϱ^{(+ ϱ + 1)}]$ . These show that, consistently, the results of [EH] are sharp....

Decomposing Baire class 1 functions into continuous functions

Saharon Shelah, Juris Steprans (1994)

Fundamenta Mathematicae

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It is shown to be consistent that every function of first Baire class can be decomposed into $ℵ_{1}$ continuous functions yet the least cardinal of a dominating family in $^{ω} ω$ is $ℵ_{2}$ . The model used in the one obtained by adding $ω_{2}$ Miller reals to a model of the Continuum Hypothesis.

On partitions of lines and space

Paul Erdös, Steve Jackson, R. Mauldin (1994)

Fundamenta Mathematicae

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We consider a set, L, of lines in $ℝ^{n}$ and a partition of L into some number of sets: $L = L_{1} \cup . . . \cup L_{p}$ . We seek a corresponding partition $ℝ^{n} = S_{1} \cup . . . \cup S_{p}$ such that each line l in $L_{i}$ meets the set $S_{i}$ in a set whose cardinality has some fixed bound, $ω_{τ}$ . We determine equivalences between the bounds on the size of the continuum, $2^{ω} \leq ω_{θ}$ , and some relationships between p, $ω_{τ}$ and $ω_{θ}$ .

A forcing construction of thin-tall Boolean algebras

Juan Martínez (1999)

Fundamenta Mathematicae

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It was proved by Juhász and Weiss that for every ordinal α with $0 < α < ω_{2}$ there is a superatomic Boolean algebra of height α and width ω. We prove that if κ is an infinite cardinal such that $κ^{< κ} = κ$ and α is an ordinal such that $0 < α < κ^{+ +}$ , then there is a cardinal-preserving partial order that forces the existence of a superatomic Boolean algebra of height α and width κ. Furthermore, iterating this forcing through all $α < κ^{+ +}$ , we obtain a notion of forcing that preserves cardinals and such that in the corresponding...

The space of ANR’s in $ℝ^{n}$

Tadeusz Dobrowolski, Leonard Rubin (1994)

Fundamenta Mathematicae

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The hyperspaces $A N R (ℝ^{n})$ and $A R (ℝ^{n})$ in $2^{ℝ^{n}} (n \geq 3)$ consisting respectively of all compact absolute neighborhood retracts and all compact absolute retracts are studied. It is shown that both have the Borel type of absolute $G_{δ σ δ}$ -spaces and that, indeed, they are not $F_{σ δ σ}$ -spaces. The main result is that $A N R (ℝ^{n})$ is an absorber for the class of all absolute $G_{δ σ δ}$ -spaces and is therefore homeomorphic to the standard model space $Ω_{3}$ of this class.

An extension of a theorem of Marcinkiewicz and Zygmund on differentiability

S. Mukhopadhyay, S. Mitra (1996)

Fundamenta Mathematicae

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Let f be a measurable function such that $Δ_{k} (x, h; f) = {O (| h |}^{λ})$ at each point x of a set E, where k is a positive integer, λ > 0 and $Δ_{k} (x, h; f)$ is the symmetric difference of f at x of order k. Marcinkiewicz and Zygmund [5] proved that if λ = k and if E is measurable then the Peano derivative $f_{(k)}$ exists a.e. on E. Here we prove that if λ > k-1 then the Peano derivative $f_{([λ])}$ exists a.e. on E and that the result is false if λ = k-1; it is further proved that if λ is any positive integer and if the approximate Peano...

Countable partitions of the sets of points and lines

James Schmerl (1999)

Fundamenta Mathematicae

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The following theorem is proved, answering a question raised by Davies in 1963. If $L_{0} \cup L_{1} \cup L_{2} \cup . . .$ is a partition of the set of lines of $ℝ^{n}$ , then there is a partition $ℝ^{n} = S_{0} \cup S_{1} \cup S_{2} \cup . . .$ such that $| ℓ \cap S_{i} | \leq 2$ whenever $ℓ \in L_{i}$ . There are generalizations to some other, higher-dimensional subspaces, improving recent results of Erdős, Jackson Mauldin.

More set-theory around the weak Freese–Nation property

Sakaé Fuchino, Lajos Soukup (1997)

Fundamenta Mathematicae

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We introduce a very weak version of the square principle which may hold even under failure of the generalized continuum hypothesis. Under this weak square principle, we give a new characterization (Theorem 10) of partial orderings with κ-Freese-Nation property (see below for the definition). The characterization is not a ZFC theorem: assuming Chang’s Conjecture for $ℵ_{ω}$ , we can find a counter-example to the characterization (Theorem 12). We then show that, in the model obtained by adding...

On character and chain conditions in images of products

Murray Bell (1998)

Fundamenta Mathematicae

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A scadic space is a Hausdorff continuous image of a product of compact scattered spaces. We complete a theorem begun by G. Chertanov that will establish that for each scadic space X, χ(X) = w(X). A ξ-adic space is a Hausdorff continuous image of a product of compact ordinal spaces. We introduce an either-or chain condition called Property $R_{λ}^{'}$ which we show is satisfied by all ξ-adic spaces. Whereas Property $R_{λ}^{'}$ is productive, we show that a weaker (but more natural) Property $R_{λ}$ is not productive....

A function space Cp(X) not linearly homeomorphic to Cp(X) × ℝ

Witold Marciszewski (1997)

Fundamenta Mathematicae

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We construct two examples of infinite spaces X such that there is no continuous linear surjection from the space of continuous functions $c_{p} (X)$ onto $c_{p} (X)$ × ℝ $. I n p a r t i c u l a r,$ cp(X) $i s n o t l i n e a r l y h o m e o m o r p h i c t o$ cp(X) $\times ℝ$ . One of these examples is compact. This answers some questions of Arkhangel’skiĭ.

Operators on C(ω^α) which do not preserve C(ω^α)

Dale Alspach (1997)

Fundamenta Mathematicae

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It is shown that if α,ζ are ordinals such that 1 ≤ ζ < α < ζω, then there is an operator from $C (ω^{ω^{α}})$ onto itself such that if Y is a subspace of $C (ω^{ω^{α}})$ which is isomorphic to $C (ω^{ω^{α}})$ , then the operator is not an isomorphism on Y. This contrasts with a result of J. Bourgain that implies that there are uncountably many ordinals α for which for any operator from $C (ω^{ω^{α}})$ onto itself there is a subspace of $C (ω^{ω^{α}})$ which is isomorphic to $C (ω^{ω^{α}})$ on which the operator is an isomorphism.

Distinguishing two partition properties of ω1

Péter Komjáth (1998)

Fundamenta Mathematicae

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It is consistent that $ω_{1} \to {(ω_{1}, (ω : 2))}^{2}$ but $ω_{1} ↛ {(ω_{1}, ω + 2)}^{2}$ .

Sumsets of Sidon sets

Imre Z. Ruzsa (1996)

Acta Arithmetica

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1. Introduction. A Sidon set is a set A of integers with the property that all the sums a+b, a,b∈ A, a≤b are distinct. A Sidon set A⊂ [1,N] can have as many as (1+o(1))√N elements, hence N/2 sums. The distribution of these sums is far from arbitrary. Erdős, Sárközy and T. Sós [1,2] established several properties of these sumsets. Among other things, in [2] they prove that A + A cannot contain an interval longer than C√N, and give an example that $N^{1 / 3}$ is possible. In [1] they show that...

Strongly meager sets and subsets of the plane

Janusz Pawlikowski (1998)

Fundamenta Mathematicae

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Let $X \subseteq 2^{w}$ . Consider the class of all Borel $F \subseteq X \times 2^{w}$ with null vertical sections $F_{x}$ , x ∈ X. We show that if for all such F and all null Z ⊆ X, $\cup_{x \in Z} F_{x}$ is null, then for all such F, $\cup_{x \in X} F_{x} \neq 2^{w}$ . The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P].

Types on stable Banach spaces

José Iovino (1998)

Fundamenta Mathematicae

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We prove a geometric characterization of Banach space stability. We show that a Banach space X is stable if and only if the following condition holds. Whenever $\hat{X}$ is an ultrapower of X and B is a ball in $\hat{X}$ , the intersection B ∩ X can be uniformly approximated by finite unions and intersections of balls in X; furthermore, the radius of these balls can be taken arbitrarily close to the radius of B, and the norm of their centers arbitrarily close to the norm of the center of B. The preceding...

Universal spaces in the theory of transfinite dimension, II

Wojciech Olszewski (1994)

Fundamenta Mathematicae

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We construct a family of spaces with “nice” structure which is universal in the class of all compact metrizable spaces of large transfinite dimension $ω_{0}$ , or, equivalently, of small transfinite dimension $ω_{0}$ ; that is, the family consists of compact metrizable spaces whose transfinite dimension is $ω_{0}$ , and every compact metrizable space with transfinite dimension $ω_{0}$ is embeddable in a space of the family. We show that the least possible cardinality of such a universal family is equal to the...