## Displaying similar documents to “A complement to the theory of equivariant finiteness obstructions”

### Normal numbers and subsets of N with given densities

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 ${D}_{\xi }\left({\Pi }_{\alpha }^{0}\right)$ iff ${D}_{X}$ is properly ${D}_{\xi }\left({\Pi }_{1+\alpha }^{0}\right)$. We also show that for every nonempty set X ⊆[0,1], ${D}_{X}$ is ${\Pi }_{3}^{0}$-hard. For each nonempty ${\Pi }_{2}^{0}$ set X ⊆ [0,1], in particular for X = x, ${D}_{X}$ is ${\Pi }_{3}^{0}$-complete. For each n ≥ 2, the collection of real numbers that are normal or simply normal to base n is ${\Pi }_{3}^{0}$-complete. Moreover,...

### Growth of the product ${\prod }_{j=1}^{n}\left(1-{x}^{{a}_{j}}\right)$

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...

### Sierpiński's hierarchy and locally Lipschitz functions

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 $\alpha <{\omega }_{1}$ then f ○ g ∈ ${B}_{\alpha }\left(Z\right)$ for every $g\in {B}_{\alpha }\left(Z\right){\cap }^{Z}I$ if and only if f is continuous on I, where ${B}_{\alpha }\left(Z\right)$ stands for the αth class in Baire’s classification of Borel measurable functions. We shall prove that for the classes ${S}_{\alpha }\left(Z\right)\left(\alpha >0\right)$ 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...

### Strongly meager sets and subsets of the plane

Fundamenta Mathematicae

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Let $X\subseteq {2}^{w}$. Consider the class of all Borel $F\subseteq X×{2}^{w}$ 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, ${\cup }_{x\in X}{F}_{x}\ne {2}^{w}$. The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P].

### Rigid ${\aleph }_{\epsilon }$ -saturated models of superstable theories

Fundamenta Mathematicae

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In a countable superstable NDOP theory, the existence of a rigid ${ﬡ}_{\epsilon }$-saturated model implies the existence of ${2}^{\lambda }$ rigid ${ﬡ}_{\epsilon }$-saturated models of power λ for every $\lambda >{2}^{{ﬡ}_{0}}$.

### How to recognize a true Σ^0_3 set

Fundamenta Mathematicae

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Let X be a Polish space, and let ${\left({A}_{p}\right)}_{p\in \omega }$ be a sequence of ${G}_{\delta }$ hereditary subsets of K(X) (the space of compact subsets of X). We give a general criterion which allows one to decide whether ${\cup }_{p\in \omega }{A}_{p}$ 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 note on strange nonchaotic attractors

Fundamenta Mathematicae

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For a class of quasiperiodically forced time-discrete dynamical systems of two variables (θ,x) ∈ ${T}^{1}×{ℝ}_{+}$ 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 ${T}^{1}×{ℝ}_{+}$. 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 ${T}^{1}×{ℝ}_{+}$ is $\Gamma ̅$, but for a residual set of points in ${T}^{1}×{ℝ}_{+}$ the omega...

### Difference functions of periodic measurable functions

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 ${\Delta }_{h}f\left(x\right)=f\left(x+h\right)-f\left(x\right)$ 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, $ℌ\left(ℱ,G\right)=H\subset ℝ/ℤ:\left(\exists f\in ℱ\phantom{\rule{4pt}{0ex}}G\right)\left(\forall h\in H\right){\Delta }_{h}f\in G$, 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...

### Partition properties of subsets of Pκλ

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 $f:{\cup }_{n<\omega }{\left[X\right]}_{\subset }^{n}\to \gamma$ with $X\subset {P}_{\kappa }\lambda$ unbounded and 1 < γ < κ there is an unbounded Y ∪ X with $|{f}^{\text{'}\text{'}}{\left[Y\right]}_{\subset }^{n}|=1$ for any n < ω.

### Strong Fubini properties of ideals

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 ${D}_{x}=y:⟨x,y⟩\in D$ are in J, then the sections ${D}^{y}=x:⟨x,y⟩\in D$ 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...

### The dimension of X^n where X is a separable metric space

Fundamenta Mathematicae

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For a separable metric space X, we consider possibilities for the sequence $S\left(X\right)={d}_{n}:n\in ℕ$ where ${d}_{n}=dim{X}^{n}$. In Section 1, a general method for producing examples is given which can be used to realize many of the possible sequences. For example, there is ${X}_{n}$ such that $S\left({X}_{n}\right)=n,n+1,n+2,...$, ${Y}_{n}$, for n >1, such that $S\left({Y}_{n}\right)=n,n+1,n+2,n+2,n+2,...$, and Z such that S(Z) = 4, 4, 6, 6, 7, 8, 9,.... In Section 2, a subset X of ${ℝ}^{2}$ is shown to exist which satisfies $1=dimX=dim{X}^{2}$ and $dim{X}^{3}=2$.

### On a discrete version of the antipodal theorem

Fundamenta Mathematicae

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The classical theorem of Borsuk and Ulam [2] says that for any continuous mapping $f:{S}^{k}\to {ℝ}^{k}$ there exists a point $x\in {S}^{k}$ 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 $in{f}_{x}||f\left(x\right)-f\left(-x\right)||$ which were previously known (as far as the author knows) only for f linear (cf. [1]).

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

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 ${\left(N+1\right)}^{-1}{\sum }_{j=0}^{N}f\left({S}^{j}x\right)\to {c}_{1}$ and ${\left(N+1\right)}^{-1}{\sum }_{j=0}^{N}f\left({T}^{j}x\right)\to {c}_{2}µ$-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 almost disjoint familes, revisited

Fundamenta Mathematicae

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The relations M(κ,λ,μ) → B [resp. B(σ)] meaning that if $A\subset {\left[\kappa \right]}^{\lambda }$ 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 $M\left({\varrho }^{\left(+\varrho +1\right)},{\varrho }^{+},\varrho \right)↛B$ [resp. $M\left({\varrho }^{\left(+\varrho +1\right)},\lambda ,\varrho \right)↛B\left({\varrho }^{+}\right)$ for all $\lambda \le {\varrho }^{\left(+\varrho +1\right)}\right]$. These show that, consistently, the results of [EH] are sharp....

### A generalization of Zeeman’s family

Fundamenta Mathematicae

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E. C. Zeeman [2] described the behaviour of the iterates of the difference equation ${x}_{n+1}=R\left({x}_{n},{x}_{n-1},...,{x}_{n-k}\right)/Q\left({x}_{n},{x}_{n-1},...,{x}_{n-k}\right)$, n ≥ k, R,Q polynomials in the case $k=1,Q={x}_{n-1}$ and $R={x}_{n}+\alpha$, ${x}_{1},{x}_{2}$ 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.

### Types on stable Banach spaces

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 $\stackrel{^}{X}$ is an ultrapower of X and B is a ball in $\stackrel{^}{X}$, 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...

### Shift spaces and attractors in noninvertible horseshoes

Fundamenta Mathematicae

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As is well known, a horseshoe map, i.e. a special injective reimbedding of the unit square ${I}^{2}$ in ${ℝ}^{2}$ (or more generally, of the cube ${I}^{m}$ in ${ℝ}^{m}$) as considered first by S. Smale [5], defines a shift dynamics on the maximal invariant subset of ${I}^{2}$ (or ${I}^{m}$). It is shown that this remains true almost surely for noninjective maps provided the contraction rate of the mapping in the stable direction is sufficiently strong, and bounds for this rate are given.

### An extension of a theorem of Marcinkiewicz and Zygmund on differentiability

Fundamenta Mathematicae

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Let f be a measurable function such that ${\Delta }_{k}\left(x,h;f\right)={O\left(|h|}^{\lambda }\right)$ at each point x of a set E, where k is a positive integer, λ > 0 and ${\Delta }_{k}\left(x,h;f\right)$ 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 ${f}_{\left(k\right)}$ exists a.e. on E. Here we prove that if λ > k-1 then the Peano derivative ${f}_{\left(\left[\lambda \right]\right)}$ 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...