Difference functions of periodic measurable functions

Tamás Keleti

Displaying similar documents to “Difference functions of periodic measurable functions”

On partitions of lines and space

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

Fundamenta Mathematicae

Similarity:

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 $ω_{θ}$ .

How to recognize a true Σ^0_3 set

Etienne Matheron (1998)

Fundamenta Mathematicae

Similarity:

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 $\sum_{3}^{0}$ .

A generalization of Zeeman’s family

Michał Sierakowski (1999)

Fundamenta Mathematicae

Similarity:

E. C. Zeeman [2] described the behaviour of the iterates of the difference equation $x_{n + 1} = R (x_{n}, x_{n - 1}, . . ., x_{n - k}) / Q (x_{n}, x_{n - 1}, . . ., x_{n - k})$ , n ≥ k, R,Q polynomials in the case $k = 1, Q = x_{n - 1}$ and $R = x_{n} + α$ , $x_{1}, x_{2}$ positive, α nonnegative. We generalize his results as well as those of Beukers and Cushman on the existence of an invariant measure in the case when R,Q are affine and k = 1. We prove that the totally invariant set remains residual when the coefficients vary.

Decomposing Baire class 1 functions into continuous functions

Saharon Shelah, Juris Steprans (1994)

Fundamenta Mathematicae

Similarity:

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.

Strongly almost disjoint familes, revisited

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

Fundamenta Mathematicae

Similarity:

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....

A forcing construction of thin-tall Boolean algebras

Juan Martínez (1999)

Fundamenta Mathematicae

Similarity:

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...

More set-theory around the weak Freese–Nation property

Sakaé Fuchino, Lajos Soukup (1997)

Fundamenta Mathematicae

Similarity:

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...

Countable partitions of the sets of points and lines

James Schmerl (1999)

Fundamenta Mathematicae

Similarity:

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.

Distinguishing two partition properties of ω1

Péter Komjáth (1998)

Fundamenta Mathematicae

Similarity:

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

Cofinal $Σ_{1}^{1}$ and $Π_{1}^{1}$ subsets of $ω^{ω}$

Gabriel Debs, Jean Saint Raymond (1999)

Fundamenta Mathematicae

Similarity:

We study properties of $\sum_{1}^{1}$ and $π_{1}^{1}$ subsets of $ω^{ω}$ that are cofinal relative to the orders ≤ (≤*) of full (eventual) domination. We apply these results to prove that the topological statement “Any compact covering mapping from a Borel space onto a Polish space is inductively perfect” is equivalent to the statement " $\forall α \in ω^{ω}, ω^{ω} \cap L (α)$ is bounded for ≤*".

The space of ANR’s in $ℝ^{n}$

Tadeusz Dobrowolski, Leonard Rubin (1994)

Fundamenta Mathematicae

Similarity:

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.

Locally constant functions

Joan Hart, Kenneth Kunen (1996)

Fundamenta Mathematicae

Similarity:

Let X be a compact Hausdorff space and M a metric space. $E_{0} (X, M)$ is the set of f ∈ C(X,M) such that there is a dense set of points x ∈ X with f constant on some neighborhood of x. We describe some general classes of X for which $E_{0} (X, M)$ is all of C(X,M). These include βℕ, any nowhere separable LOTS, and any X such that forcing with the open subsets of X does not add reals. In the case where M is a Banach space, we discuss the properties of $E_{0} (X, M)$ as a normed linear space. We also build three first countable...

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

Similarity:

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...

Types on stable Banach spaces

José Iovino (1998)

Fundamenta Mathematicae

Similarity:

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...

Strongly meager sets and subsets of the plane

Janusz Pawlikowski (1998)

Fundamenta Mathematicae

Similarity:

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].

On character and chain conditions in images of products

Murray Bell (1998)

Fundamenta Mathematicae

Similarity:

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....

Universal spaces in the theory of transfinite dimension, II

Wojciech Olszewski (1994)

Fundamenta Mathematicae

Similarity:

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...

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

Witold Marciszewski (1997)

Fundamenta Mathematicae

Similarity:

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.

An extension of a theorem of Marcinkiewicz and Zygmund on differentiability

S. Mukhopadhyay, S. Mitra (1996)

Fundamenta Mathematicae

Similarity:

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...