On a discrete version of the antipodal theorem

Krzysztof Oleszkiewicz

Displaying similar documents to “On a discrete version of the antipodal theorem”

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,...

Ergodic averages and free $ℤ^{2}$ actions

Zoltán Buczolich (1999)

Fundamenta Mathematicae

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If the ergodic transformations S, T generate a free $ℤ^{2}$ action on a finite non-atomic measure space (X,S,µ) then for any $c_{1}, c_{2} \in ℝ$ there exists a measurable function f on X for which ${(N + 1)}^{- 1} \sum_{j = 0}^{N} f (S^{j} x) \to c_{1}$ and ${(N + 1)}^{- 1} \sum_{j = 0}^{N} f (T^{j} x) \to c_{2} µ$ -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 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].

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...

A note on strange nonchaotic attractors

Gerhard Keller (1996)

Fundamenta Mathematicae

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For a class of quasiperiodically forced time-discrete dynamical systems of two variables (θ,x) ∈ $T^{1} \times ℝ_{+}$ with nonpositive Lyapunov exponents we prove the existence of an attractor Γ̅ with the following properties: 1. Γ̅ is the closure of the graph of a function x = ϕ(θ). It attracts Lebesgue-a.e. starting point in $T^{1} \times ℝ_{+}$ . The set θ:ϕ(θ) ≠ 0 is meager but has full 1-dimensional Lebesgue measure. 2. The omega-limit of Lebesgue-a.e point in $T^{1} \times ℝ_{+}$ is $Γ ̅$ , but for a residual set of points in $T^{1} \times ℝ_{+}$ the omega...

Growth of the product $\prod_{j = 1}^{n} (1 - x^{a_{j}})$

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

Acta Arithmetica

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We estimate the maximum of $\prod_{j = 1}^{n} | 1 - x^{a_{j}} |$ 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...

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....

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 $A : 𝕆_{G} \to 𝔸$ 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 $E x t^{n - 1} (A, A \otimes A)$ . Then the case $G = ℤ_{p^{k}}$ leads to an example of...

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 $f : \cup_{n < ω} {[X]}_{\subset}^{n} \to γ$ with $X \subset P_{κ} λ$ unbounded and 1 < γ < κ there is an unbounded Y ∪ X with $| f^{''} {[Y]}_{\subset}^{n} | = 1$ for any n < ω.

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...

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.

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...

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

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 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....

$Σ$ -products of paracompact Čech-scattered spaces

Hidenori Tanaka (2006)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we shall discuss $Σ$ -products of paracompact Čech-scattered spaces and show the following: (1) Let $Σ$ be a $Σ$ -product of paracompact Čech-scattered spaces. If $Σ$ has countable tightness, then it is collectionwise normal. (2) If $Σ$ is a $Σ$ -product of first countable, paracompact (subparacompact) Čech-scattered spaces, then it is shrinking (subshrinking).