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,...

We parametrize Cichoń’s diagram and show how cardinals from Cichoń’s diagram yield classes of small sets of reals. For instance, we show that there exist subsets N and M of $w^w\times 2^w$ and continuous functions $e,f:w^w\to w^w$ such that  • N is $G_{\delta }$ and $N_x:x\in w^w$, the collection of all vertical sections of N, is a basis for the ideal of measure zero subsets of $2^w$;  • M is $F_{\sigma }$ and $M_x:x\in w^w$ is a basis for the ideal of meager subsets of $2^w$;  •$\forall x,y{N}_{e\left(x\right)}\subseteq N_y\Rightarrow M_x\subseteq M_{f\left(y\right)}$. From this we derive that for a separable metric space X,  •if for all Borel (resp. $G_{\delta }$) sets...

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 $Ex{t}^{n-1}(A,A\otimes A)$. Then the case $G={\mathbb{Z}}_{p^k}$ leads to an example of...

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, ${\cup }_{x\in X}{F}_x\ne 2^w$. The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P].

We prove the existence of Carathéodory selections and representations of a closed convex valued, lower Carathéodory multifunction from a set A in $A\left(ℰ\otimes ℬ\right(X\left)\right)$ 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...

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...

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 $in{f}_x\left|\right|f\left(x\right)-f(-x)\left|\right|$ which were previously known (as far as the author knows) only for f linear (cf. [1]).

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 $S_{\alpha }\left(Z\right)(\alpha >0)$ 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...

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 $\Gamma ̅$, but for a residual set of points in $T^1\times {ℝ}_{+}$ the omega...

Using Tsirelson’s well-known example of a Banach space which does not contain a copy of $c_0$ or $l_p$, for p ≥ 1, we construct a simple Borel ideal $I_T$ such that the Borel cardinalities of the quotient spaces $P\left(\mathbb{N}\right)/I_T$ and $P\left(\mathbb{N}\right)/I_0$ are incomparable, where $I_0$ is the summable ideal of all sets A ⊆ ℕ such that ${\sum }_{n\in A}1/(n+1)<\infty$. This disproves a “trichotomy” conjecture for Borel ideals proposed by Kechris and Mazur.

If the ergodic transformations S, T generate a free ${\mathbb{Z}}^2$ 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 ${(N+1)}^{-1}{\sum }_{j=0}^{N}f\left(T^{j}x\right)\to c_2\mu$-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.

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 ${\Delta }_{h}f\left(x\right)=f(x+h)-f\left(x\right)$ 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, $ℌ(ℱ,G)=H\subset ℝ/\mathbb{Z}:\left(\exists f\in ℱG\right)\left(\forall h\in H\right){\Delta }_{h}f\in G$, 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 $𝕋=ℝ/\mathbb{Z}$ that are invariant for changes on null-sets...

For a separable metric space X, we consider possibilities for the sequence $S\left(X\right)=d_n:n\in \mathbb{N}$ 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 $dim{X}^3=2$.

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 $R=x_n+\alpha$, $x_1,x_2$ 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.