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,...

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].

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]).

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...

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.

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 $M({\varrho }^{(+\varrho +1)},{\varrho }^{+},\varrho )\nrightarrow B$ [resp. $M({\varrho }^{(+\varrho +1)},\lambda ,\varrho )\nrightarrow B\left({\varrho }^{+}\right)$ for all $\lambda \le {\varrho }^{(+\varrho +1)}]$. These show that, consistently, the results of [EH] are sharp....

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...

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 $DX,X=(X,n)\in P$, turns out to be of the same weak homotopy type as an appropriately defined functional dual $\overline{{\left(S^0\right)}^X}$ (Corollary 4.9). Sometimes the functional object $\overline{X^Y}$ is of the same weak homotopy type as the “real” function space $X^Y$ (§5).

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...

The arithmetic function $r_k\left(n\right)$ 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 $r_k\left(n\right)$ leads in a natural way to a certain error term $P_{{}_k}\left(t\right)$ which is known to be $O\left(t^{1/4}\right)$ in mean-square. In this article it is proved that $P_{₃}\left(t\right)=\Omega ₊\left(t^{1/4}{\left(loglogt\right)}^{1/4}\right)$ 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...

Let α be an ergodic rotation of the d-torus ${𝕋}^d={ℝ}^d/{\mathbb{Z}}^d$. For any piecewise smooth function $f:{𝕋}^d\to ℝ$ with sufficiently regular pieces the unitary operator Vh(x) = exp(2π if(x))h(x + α) acting on $L^2\left({𝕋}^d\right)$ is shown to have a continuous non-Dirichlet spectrum if the gradient of f has nonzero integral. In particular, the resulting skew product $S_f:{𝕋}^{d+1}\to {𝕋}^{d+1}$ must be ergodic. If in addition α is sufficiently well approximated by rational vectors and f is represented by a linear function with noninteger coefficients then the spectrum...

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...

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

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.

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.

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...

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...

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 $R_{\lambda }^{\text{'}}$ which we show is satisfied by all ξ-adic spaces. Whereas Property $R_{\lambda }^{\text{'}}$ is productive, we show that a weaker (but more natural) Property $R_{\lambda }$ is not productive....

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 $dim{X}^3=2$.