Displaying similar documents to “A note on strange nonchaotic attractors”

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,...

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].

Growth of the product j = 1 n ( 1 - x a j )

J. P. Bell, P. B. Borwein, L. B. Richmond (1998)

Acta Arithmetica

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We estimate the maximum of j = 1 n | 1 - x a j | on the unit circle where 1 ≤ a₁ ≤ a₂ ≤ ... is a sequence of integers. We show that when a j is j k or when a j is a quadratic in j that takes on positive integer values, the maximum grows as exp(cn), where c is a positive constant. This complements results of Sudler and Wright that show exponential growth when a j is j.    In contrast we show, under fairly general conditions, that the maximum is less than 2 n / n r , where r is an arbitrary positive number. One consequence...

On a discrete version of the antipodal theorem

Krzysztof Oleszkiewicz (1996)

Fundamenta Mathematicae

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The classical theorem of Borsuk and Ulam [2] says that for any continuous mapping f : S k k there exists a point x S k such that f(-x) = f(x). In this note a discrete version of the antipodal theorem is proved in which S k is replaced by the set of vertices of a high-dimensional cube equipped with Hamming’s metric. In place of equality we obtain some optimal estimates of i n f x | | f ( x ) - f ( - x ) | | which were previously known (as far as the author knows) only for f linear (cf. [1]).

Ergodic averages and free 2 actions

Zoltán Buczolich (1999)

Fundamenta Mathematicae

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If the ergodic transformations S, T generate a free 2 action on a finite non-atomic measure space (X,S,µ) then for any c 1 , c 2 there exists a measurable function f on X for which ( N + 1 ) - 1 j = 0 N f ( S j x ) c 1 and ( N + 1 ) - 1 j = 0 N f ( T j x ) c 2 µ -almost everywhere as N → ∞. In the special case when S, T are rationally independent rotations of the circle this result answers a question of M. Laczkovich.

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 [ 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 ϱ ( + ϱ + 1 ) ] . These show that, consistently, the results of [EH] are sharp....

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...

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.

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...

Co-H-structures on equivariant Moore spaces

Martin Arkowitz, Marek Golasiński (1994)

Fundamenta Mathematicae

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Let G be a finite group, 𝕆 G the category of canonical orbits of G and A : 𝕆 G 𝔸 b a contravariant functor to the category of abelian groups. We investigate the set of G-homotopy classes of comultiplications of a Moore G-space of type (A,n) where n ≥ 2 and prove that if such a Moore G-space X is a cogroup, then it has a unique comultiplication if dim X < 2n - 1. If dim X = 2n-1, then the set of comultiplications of X is in one-one correspondence with E x t n - 1 ( A , A A ) . Then the case G = p k leads to an example of...

Sierpiński's hierarchy and locally Lipschitz functions

Michał Morayne (1995)

Fundamenta Mathematicae

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Let Z be an uncountable Polish space. It is a classical result that if I ⊆ ℝ is any interval (proper or not), f: I → ℝ and α < ω 1 then f ○ g ∈ B α ( Z ) for every g B α ( Z ) Z I if and only if f is continuous on I, where B α ( Z ) stands for the αth class in Baire’s classification of Borel measurable functions. We shall prove that for the classes S α ( Z ) ( α > 0 ) in Sierpiński’s classification of Borel measurable functions the analogous result holds where the condition that f is continuous is replaced by the condition that f is locally...

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.

Borel partitions of unity and lower Carathéodory multifunctions

S. Srivastava (1995)

Fundamenta Mathematicae

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We prove the existence of Carathéodory selections and representations of a closed convex valued, lower Carathéodory multifunction from a set A in A ( ( X ) ) into a separable Banach space Y, where ℰ is a sub-σ-field of the Borel σ-field ℬ(E) of a Polish space E, X is a Polish space and A is the Suslin operation. As applications we obtain random versions of results on extensions of continuous functions and fixed points of multifunctions. Such results are useful in the study of random differential...

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 .

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 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....

Strong Fubini properties of ideals

Ireneusz Recław, Piotr Zakrzewski (1999)

Fundamenta Mathematicae

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 Let I and J be σ-ideals on Polish spaces X and Y, respectively. We say that the pair ⟨I,J⟩ has the Strong Fubini Property (SFP) if for every set D ⊆ X× Y with measurable sections, if all its sections D x = y : x , y D are in J, then the sections D y = x : x , y D are in I for every y outside a set from J (“measurable" means being a member of the σ-algebra of Borel sets modulo sets from the respective σ-ideal). We study the question of which pairs of σ-ideals have the Strong Fubini Property. Since CH excludes this...

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 .