Displaying similar documents to “Strongly almost disjoint familes, revisited”

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...

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 ( 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...

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 n and a partition of L into some number of sets: L = L 1 . . . L p . We seek a corresponding partition n = S 1 . . . 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 ω θ , and some relationships between p, ω τ and ω θ .

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, ( , G ) = H / : ( f G ) ( h H ) Δ h f 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 = / that are invariant for changes on null-sets...

A generalization of Zeeman’s family

Michał Sierakowski (1999)

Fundamenta Mathematicae

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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

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

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 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 p ω A p is a true 3 0 subset of K(X). We apply this criterion to show that several natural families of thin sets from harmonic analysis are true 3 0 .

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...

Strongly meager sets and subsets of the plane

Janusz Pawlikowski (1998)

Fundamenta Mathematicae

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

Partition properties of subsets of Pκλ

Masahiro Shioya (1999)

Fundamenta Mathematicae

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Let κ > ω be a regular cardinal and λ > κ a cardinal. The following partition property is shown to be consistent relative to a supercompact cardinal: For any f : n < ω [ X ] n γ with X P κ λ unbounded and 1 < γ < κ there is an unbounded Y ∪ X with | f ' ' [ Y ] n | = 1 for any n < ω.

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 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....

Normal numbers and subsets of N with given densities

Haseo Ki, Tom Linton (1994)

Fundamenta Mathematicae

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For X ⊆ [0,1], let D X denote the collection of subsets of ℕ whose densities lie in X. Given the exact location of X in the Borel or difference hierarchy, we exhibit the exact location of D X . For α ≥ 3, X is properly D ξ ( Π α 0 ) iff D X is properly D ξ ( Π 1 + α 0 ) . We also show that for every nonempty set X ⊆[0,1], D X is Π 3 0 -hard. For each nonempty Π 2 0 set X ⊆ [0,1], in particular for X = x, D X is Π 3 0 -complete. For each n ≥ 2, the collection of real numbers that are normal or simply normal to base n is Π 3 0 -complete. Moreover,...

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.

The space of ANR’s in n

Tadeusz Dobrowolski, Leonard Rubin (1994)

Fundamenta Mathematicae

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The hyperspaces A N R ( n ) and A R ( n ) in 2 n ( n 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 ( 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.

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) × . One of these examples is compact. This answers some questions of Arkhangel’skiĭ.

A functional S-dual in a strong shape category

Friedrich Bauer (1997)

Fundamenta Mathematicae

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In the S-category P (with compact-open strong shape mappings, cf. §1, instead of continuous mappings, and arbitrary finite-dimensional separable metrizable spaces instead of finite polyhedra) there exists according to [1], [2] an S-duality. The S-dual D X , X = ( X , n ) P , turns out to be of the same weak homotopy type as an appropriately defined functional dual ( S 0 ) X ¯ (Corollary 4.9). Sometimes the functional object X Y ¯ is of the same weak homotopy type as the “real” function space X Y (§5).

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 L 1 L 2 . . . is a partition of the set of lines of n , then there is a partition n = S 0 S 1 S 2 . . . such that | S i | 2 whenever L i . There are generalizations to some other, higher-dimensional subspaces, improving recent results of Erdős, Jackson Mauldin.